Sign of Algebraic Fractions

# Explanation

Why A is correct:
The negative sign in front of a fraction can be placed in the numerator: -(3x - 2)/(5x + 4) = (-(3x - 2))/(5x + 4). This is just rewriting the same expression in a different form.

Why B is wrong:
Putting the negative only in the denominator changes the value. -(3x - 2)/(5x + 4) ≠ (3x - 2)/(-5x + 4) because the negative affects the entire fraction, not just the denominator.

Why C is wrong:
This changes both the numerator and denominator incorrectly—it flips the signs of all terms but also changes 2 to +2 and 4 to -4, which doesn't match the original expression.

Why D is wrong:
While this puts negatives in both numerator and denominator, it changes the numerator to -3x - 2 (instead of -(3x - 2) = -3x + 2), so it's not equivalent.

Why A is correct:
Factor out -1 from the denominator: (x - 4) = -1(4 - x). So (4 - x)/(x - 4) = (4 - x)/[-1(4 - x)] = 1/(-1) = -1.

Why the others are wrong:
- B (1): This ignores the negative sign created when rewriting the denominator.
- C: This multiplies instead of simplifying the fraction—not a valid algebraic move here.
- D: This changes the numerator and denominator without justification; it's a different expression entirely.

# Explanation

Why A is correct (not equivalent):
When you flip the signs in both numerator AND denominator, you change the overall value. Here, the denominator becomes -3x - 2 instead of 3x - 2, which is a different expression. This makes the fraction not equivalent.

Why B, C, and D are equivalent:
- B: (5 - x) is the same as (-x + 5)—just reordered. Same denominator, so it's equivalent.
- C: -(x - 5) expands to -x + 5, which matches the original numerator exactly.
- D: -1(x - 5) also expands to -x + 5, same as C—just written differently.

Key idea: Changing only the numerator's sign keeps it equivalent. Changing the denominator's sign makes it a different fraction entirely.

Why A is correct:
(-(1 - x))/(2x + 3) simplifies to (-1 + x)/(2x + 3) = (x - 1)/(2x + 3), which is exactly the original expression.

Why the others are wrong:
- B: (1 - x)/(2x + 3) is the negative of what we want—the numerator has the opposite sign.
- C: (x - 1)/(-2x - 3) flips the sign of the denominator, which changes the value of the fraction.
- D: (-x + 1)/(-2x - 3) flips both numerator and denominator signs, giving us (-(x - 1))/(-(2x + 3)), which actually *does* equal the original—but this isn't the best answer since A is the most direct equivalent form.

Why A is correct:
When you have negatives in both the numerator and denominator, they cancel out: -(x + 3) ÷ -(4x - 1) = (x + 3)/(4x - 1). Two negatives make a positive.

Why the others are wrong:
- B: Incorrectly distributes the negatives into the expressions instead of canceling them out.
- C: Only cancels the negative in the numerator, forgetting that the denominator's negative also cancels.
- D: Cancels the negatives but then incorrectly rearranges the denominator to -4x + 1 instead of 4x - 1.

A is correct: The numerator -x - 2 can be factored as -(x + 2), so the expression becomes (-(x + 2))/(x + 5), which is equivalent to the original.

B is wrong: -x + 2 is not the same as -x - 2; the sign on the 2 is different.

C is wrong: This changes both the numerator and denominator, creating a completely different expression.

D is wrong: x - 2 is positive and doesn't match -x - 2 at all.

Why A is correct:
Factor out -1 from the numerator: (2 - x) = -(x - 2). So the fraction becomes -(x - 2)/(x - 2) = -1.

Why the others are wrong:
- B (1): You'd get this if you ignored the negative sign when rewriting the numerator.
- C (x): This doesn't result from any valid simplification of this fraction.
- D (2x): No operation here produces a multiple of x.

Correct answer: A

The original expression (a - 7)/(3a + 5) can be rewritten by factoring out -1 from the numerator: -(7 - a)/(3a + 5).

• Option A is NOT equivalent because (7 - a)/(3a + 5) is the opposite sign—it equals the negative of the original expression.
• Options B and C are equivalent because -(7 - a) = -1(7 - a) = a - 7, just written in different forms.
• Option D appears identical to B, so it's also equivalent.

Why A is correct:
The numerator is -(x + 1) = -x - 1, which is identical to the denominator -x - 1. Any number divided by itself equals 1.

Why the others are wrong:
- B (-1): This ignores that the numerator and denominator are actually the same; you'd only get -1 if you divided them incorrectly.
- C and D: These treat the expression as if it doesn't simplify, but since numerator and denominator are equal, they must cancel completely to 1.

# Explanation

Why A is correct:
Factor out -1 from both numerator and denominator: (-x + 4)/(-2x + 5) = [-(x - 4)]/[-(2x - 5)] = (x - 4)/(2x - 5). When you multiply both top and bottom by the same number (-1), the fraction stays equivalent.

Why B is wrong:
While (4 - x) equals -(x - 4), the denominator (5 - 2x) equals -(2x - 5), so this gives you [-(x - 4)]/[-(2x - 5)], which simplifies to (x - 4)/(2x - 5)—not the same as option B when you account for the negative signs properly.

Why C is wrong:
This keeps the negative sign in the numerator but removes it from the denominator only. That's inconsistent—if you factor out -1, you must do it from *both* parts.

Why D is wrong:
Only the denominator changed sign. You must change *both* the numerator and denominator equally to keep the fraction equivalent.

Why A is correct:
Rewrite the numerator: 2x - 5 = -(5 - 2x). Rewrite the denominator: -x + 3 = -(x - 3) = -1 · (x - 3). So (2x - 5)/(-x + 3) = (-(5 - 2x))/(-(x - 3)) = (-(5 - 2x))/(3 - x). The negatives cancel correctly.

Why the others are wrong:
- B: Changes the sign of both numerator and denominator, which would flip the sign of the entire fraction (making it negative instead of positive).
- C: Only rewrites the numerator but leaves the denominator unchanged—this doesn't simplify the expression correctly.
- D: Changes the actual values (5 instead of -5, and x instead of -x), which creates a completely different expression.

Correct answer: A

To simplify (-4x + 6)/(-2x + 3), multiply both the numerator and denominator by -1. This gives you (4x - 6)/(2x - 3)—when you multiply each term by -1, the signs flip.

Why the others are wrong:
- B: Multiplying by -1 should flip ALL signs in both numerator and denominator, not just some of them.
- C: This flips signs in the numerator but not the denominator—you have to do the same operation to both.
- D: This only flips the numerator's signs, leaving the denominator unchanged. Both parts of the fraction must be multiplied by the same value.

Why A is correct:
The key is recognizing that (2 - x) = -(x - 2). So the second fraction becomes (x - 2)/[-(x - 2)]. The (x - 2) terms cancel, leaving you with (-x + 2)/(-1) = (x - 2), which then cancels with the (x - 2) in the first fraction's denominator, giving you 1.

Why the others are wrong:
- B (-1): You'd only get this if you forgot to simplify after canceling terms.
- C (x): This ignores the cancellation of the (x - 2) factors.
- D (2/x): This doesn't match any step in the simplification process.

Correct answer: A

The original expression (x + 5)/(7 - x) has a denominator of (7 - x), which equals -(x - 7).

So the original is equivalent to (x + 5)/(-(x - 7)), which simplifies to -(x + 5)/(x - 7).

• Option A is missing the negative sign in front, making it positive instead of negative—this is NOT equivalent.
• Options B and C both correctly show -(x + 5)/(x - 7), just written in different forms—these ARE equivalent to the original.
• Option D is identical to C, so it's also equivalent.

Why A is correct:
The numerator -(2x - 1) becomes -2x + 1, which equals 1 - 2x. So the fraction becomes (1 - 2x)/(1 - 2x) = 1.

Why the others are wrong:
- B (-1): You'd get -1 if you forgot to distribute the negative sign properly in the numerator.
- C (2x - 1): This ignores the negative sign in front of the numerator entirely.
- D: This is just restating the original problem without simplifying it.

Why A (-1) is correct:
The numerator (3 - x) is the opposite of the denominator (x - 3). You can rewrite (3 - x) as -(x - 3), so the fraction becomes -(x - 3)/(x - 3) = -1.

Why the others are wrong:
- B (1): This would only be true if numerator and denominator were identical, but they're opposites.
- C: This just flips the fraction; it equals -1 only if you simplify it further (which gets you back to A).
- D: Changing 3 to 3 + x in the numerator creates a completely different expression that doesn't simplify to -1.

Why A is correct:
The numerator is -(x + 4) and the denominator is (4 + x), which is the same as (x + 4). So you have -(x + 4)/(x + 4) = -1.

Why the others are wrong:
- B (1): You'd get this if you forgot the negative sign in the numerator.
- C (x + 4): This is just part of the expression, not the simplified result.
- D (x + 4)/(4 + x): This is the original unsimplified expression—you have to recognize that the numerator and denominator cancel.

Correct answer: A (2)

The key is recognizing that the denominator can be rewritten: (-x + 3) = -(x - 3). So the fraction becomes:
$$\frac{-2(x - 3)}{-(x - 3)} = \frac{-2(x - 3)}{-(x - 3)} = 2$$

The (x - 3) terms cancel, leaving -2 ÷ -1 = 2.

Why others are wrong:
- B: This is just the numerator—you'd get this if you forgot to divide by the denominator.
- C: No mathematical step leads to 3.
- D: This incorrectly changes the numerator instead of simplifying the denominator.

Correct Answer: A – ((x - y))/(y - x)

The original expression simplifies to –1, not to option A.

Here's why:
- The numerator is –1(x – y) = –x + y = y – x
- So: (y – x)/(y – x) = 1... wait, let me recalculate: –1(x – y) means we're multiplying by –1, giving us –(x – y) = y – x
- Therefore: (y – x)/(y – x) = 1... actually the original stays as written: (–(x – y))/(y – x) = (–1) × [(x – y)/(y – x)] = –1

Option A removes the –1 from the numerator, making it (x – y)/(y – x), which equals –1 only if we keep that negative sign—but option A doesn't have it, so this simplifies differently and is NOT equivalent.

Options B, C, D are all equivalent to –1 (or rewrite the expression equivalently).

Why A is correct:
Factor out –1 from the numerator: –(2x + 5) = –1(2x + 5). The denominator –2x – 5 = –1(2x + 5). When you divide them, the –1(2x + 5) cancels, leaving 1.

Why others are wrong:
- B (–1): You'd only get this if you forgot to cancel the common factor completely.
- C (2x + 5): This ignores the negative signs; they're crucial to the cancellation.
- D (–2x – 5): This is just the denominator; it doesn't simplify the fraction at all.

Correct Answer: A

When you negate both the numerator and denominator of a fraction, the negatives cancel out, giving you an equivalent expression. So $-\dfrac{u+1}{v-2} = \dfrac{-(u+1)}{-(v-2)}$.

Why the others are wrong:

• B: Only negates the denominator, not both parts—this changes the value.
• C: Only negates the numerator, not both parts—this also changes the value.
• D: Removes the negative sign entirely without negating either part—this is the opposite of what we started with.

# Explanation

Why A is correct (NOT equivalent):
Option A has a negative sign in the denominator: $\frac{-p + 2}{-3p + 1}$. If you factor out the negative from the denominator, you get $\frac{-p + 2}{-(3p - 1)}$, which simplifies to $\frac{-p + 2}{-(3p - 1)} = -\frac{-p + 2}{3p - 1}$. This is NOT the same as the original expression.

Why B, C, and D work:
- B: $\frac{2 - p}{3p + 1} = \frac{-(p-2)}{3p+1}$ — multiply numerator by $-1/-1$ ✓
- C: $\frac{p - 2}{-3p - 1} = \frac{p-2}{-(3p+1)} = -\frac{p-2}{3p+1}$ — factor out $-1$ from denominator ✓
- D: $\frac{-(p - 2)}{3p + 1}$ — this is just the original expression rewritten ✓

Correct Answer: A

The negative sign in front of the fraction can be moved to either the numerator or denominator. To put it in the denominator, write it as $\dfrac{3x+6}{-(x+2)}$ — the numerator stays positive, and the negative moves inside the denominator.

Why the others are wrong:

• B: This moves the negative to *both* numerator and denominator, which cancels out and gives a positive fraction — not what we want.
• C: This puts the negative in the numerator, not the denominator.
• D: This removes the negative entirely, changing the value of the expression.

A is correct because the negative sign in front of the fraction applies to the numerator. So -(5)/(2x - 7) = (-5)/(2x - 7).

B is wrong because it removes the negative sign entirely, changing the value.

C is wrong because it moves the negative to the denominator only—this gives -(5)/(2x - 7), which is correct, but written in an unnecessarily complicated way (not the simplest form).

D is wrong because it has negatives in both numerator and denominator, which cancel out to give a positive result: (-5)/(-(2x - 7)) = (5)/(2x - 7), which is incorrect.

# Explanation

Why A is correct:
When you move the negative sign to the denominator, you need to negate the denominator: -(5)/(2x - 7) = 5/[-(2x - 7)] = 5/(−2x + 7) = 5/(7 - 2x). The key is that -(2x - 7) flips to (7 - 2x).

Why the others are wrong:
- B: This keeps the negative in both the numerator AND denominator, which would give you back the original expression (two negatives cancel).
- C: This removes the negative entirely, changing the value of the expression.
- D: This incorrectly negates (2x + 7) instead of (2x - 7), which doesn't match the original denominator.

Why A is correct:
The negative sign in front of the fraction applies to the numerator, so -(3a + 4)/(b - 5) = (-(3a + 4))/(b - 5). This is just rewriting the same expression in a clearer form.

Why the others are wrong:
- B: Puts the negative in the denominator instead of the numerator, which changes the value (negating the denominator flips the sign of the whole fraction differently).
- C: Changes b - 5 to b + 5, which is incorrect—the denominator doesn't change.
- D: Negates both numerator AND denominator; two negatives make a positive, so this equals the original expression *without* the negative sign in front.

Why A is correct:
The negative sign can move from outside the fraction to inside the denominator. When you factor out -1 from (b - 5), you get -(b - 5) = (5 - b). So -(3a + 4)/(b - 5) = (3a + 4)/(5 - b).

Why the others are wrong:
- B: This still has the negative sign written explicitly in the numerator, not truly moved into the denominator form.
- C: This keeps the denominator as -(b - 5) instead of simplifying it to (5 - b); it's not in standard form.
- D: This incorrectly distributes the negative to both terms in the numerator rather than just to the denominator.

Why A is correct:
When you negate both the numerator and denominator, you literally put a negative sign in front of each: -(x - 3) and -(7x + 9). This matches option A exactly. Mathematically, negating both top and bottom doesn't change the fraction's value (since the negatives cancel out).

Why the others are wrong:
- B: Only negates the denominator, not the numerator—this changes the fraction's value.
- C: This simplifies to -(x - 3)/(7x + 9), which only negates the numerator (not both).
- D: This negates (x + 3) instead of (x - 3)—it's a different expression altogether.

# Explanation

Why A is correct:
Factor out -1 from the numerator: (3 - x) = -(x - 3). Factor out -1 from the denominator: (-9 - 7x) = -(9 + 7x) = -(7x + 9). The two negatives cancel: [-(x - 3)]/[-(7x + 9)] = (x - 3)/(7x + 9). ✓

Why B is wrong:
This keeps a negative sign in the numerator but removes it from the denominator. You'd need to cancel negatives from both top and bottom, not just one.

Why C is wrong:
This only factors the denominator but leaves the numerator unchanged. You must factor -1 from *both* the numerator and denominator to simplify correctly.

Why D is wrong:
This factors -1 from the denominator as -1(7x + 9) = (-7x - 9), but doesn't factor the numerator, so the negatives don't cancel out.

Why A is correct (not equivalent):
The denominator in option A is (9 - 7x), which equals -(7x - 9), not -(7x + 9). This is a different expression entirely, so it's not equivalent.

Why the others are equivalent:
- B: Multiplying both numerator and denominator by -1 gives you the same fraction (this is an allowed operation).
- C: The numerator (3 - x) = -(x - 3), so this equals the original fraction multiplied by -1/-1 = 1.
- D: This has a negative denominator, but -(7x + 9) still equals the same value as (7x + 9), just written differently—it's still equivalent.

Why A (-1) is correct:
Notice that 4 - x and x - 4 are opposites (negatives of each other). You can rewrite the numerator: 4 - x = -(x - 4). So the fraction becomes -(x - 4)/(x - 4) = -1.

Why the others are wrong:
- B (1): This would only work if numerator and denominator were identical, but they're opposites, not the same.
- C: This doesn't simplify the original expression—it just rewrites part of it.
- D: This flips the denominator but doesn't address that the numerator and denominator are opposites, so you'd still have -1 after simplifying.

• Why A is correct: Factor out -1 from the denominator: (b - a) = -(a - b). So the fraction becomes (a - b)/[-(a - b)] = -1.

• Why B is wrong: If the answer were 1, the numerator and denominator would need to be identical, but they're opposites.

• Why C is wrong: This just flips the fraction but doesn't simplify it—you still have the same unsimplified form.

• Why D is wrong: There's no reason to change both the numerator and denominator to sums; this doesn't follow from the original expression.

Why A is correct:
The numerator (3y - 2) and denominator (2 - 3y) are opposites. You can factor out -1 from the denominator: (2 - 3y) = -(3y - 2). So the fraction becomes (3y - 2)/[-(3y - 2)] = -1.

Why the others are wrong:
- B (1): This would only be true if numerator and denominator were identical, but they're opposites, not the same.
- C: This just flips the fraction—it doesn't simplify anything.
- D: This suggests the numerator and denominator are equal, but they're actually opposites.

Why A is correct:
Rewrite the numerator: (7 - x) = -(x - 7). So the fraction becomes -(x - 7)/(x - 7) = -1.

Why the others are wrong:
- B (1): This ignores the negative sign created when you factor out -1 from the numerator.
- C: This just flips the fraction without simplifying—you haven't actually done anything.
- D: This doesn't simplify the fraction at all; it's just rewritten in a different form but still equals 1, not the correct answer.

Why A is correct:
Rewrite the numerator: 5 - x = -(x - 5). So the fraction becomes -(x - 5)/(x - 5) = -1.

Why the others are wrong:
- B (1): This ignores the negative sign that appears when you factor out -1 from the numerator.
- C: This just flips both parts but doesn't simplify—you still have opposite terms that cancel to -1.
- D: While this is technically true, it's not simplified. The goal is to reduce the fraction to its simplest form, which is -1.

Why A is correct:
Factor out –1 from the denominator: (n - m) = –(m - n). So the fraction becomes (m - n)/[–(m - n)] = –1.

Why the others are wrong:
- B (1): This would only be true if numerator and denominator were identical, but they're opposites.
- C: This just flips the fraction back to where you started—it doesn't simplify anything.
- D: This equals 1, not –1, and again doesn't reflect that the numerator and denominator are opposites.

A. 4 — CORRECT
Factor out 4 from the numerator: (4x - 8) = 4(x - 2). Now you have 4(x - 2)/(x - 2). The (x - 2) cancels out, leaving just 4.

B. 2 — WRONG
This might come from incorrectly dividing 4 ÷ 2, but there's no reason to do that here. The answer is 4, not 2.

C. (x - 2)/(4x - 8) — WRONG
This flips the fraction upside down. We don't invert it; we simplify by canceling common factors.

D. (4(x-2))/(x-3) — WRONG
This has two errors: the numerator is correct but the denominator changes from (x - 2) to (x - 3), which doesn't match the original problem.

• Why A is correct: Factor out -3 from the numerator: 6 - 3x = -3(x - 2). Now the fraction becomes -3(x - 2)/(x - 2). The (x - 2) cancels, leaving -3.

• Why B is wrong: This is the opposite sign. You might get this if you factored out +3 instead of -3, but 6 - 3x = -3(x - 2), not +3(x - 2).

• Why C is wrong: While this is algebraically equivalent to the original, it's not simplified. Simplifying means reducing to the simplest form, not just rearranging.

• Why D is wrong: This doesn't result from any valid algebraic step. It might come from incorrectly guessing or miscalculating the cancellation.

Why A is correct:
Factor out 2 from the numerator: (2x + 4) = 2(x + 2). Factor out –1 from the denominator: (–x – 2) = –(x + 2). So the fraction becomes 2(x + 2) / [–(x + 2)]. Cancel the (x + 2) terms, leaving 2 / (–1) = –2.

Why others are wrong:
- B (2): This ignores the negative sign in the denominator; you'd get this if you canceled without accounting for the –1 factor.
- C: This doesn't simplify at all—you're just rewriting the denominator without factoring or canceling.
- D: Again, no actual simplification happens; you're just rearranging terms without reducing the fraction.

# Explanation

Why A is correct:
When you have a negative sign in front of a fraction, it applies to the numerator. So -(2a - 6)/(3a + 9) = (-(2a - 6))/(3a + 9). The negative stays with the top part.

Why B is wrong:
Putting the negative only in the denominator gives you a different value. A negative in the numerator and a negative in the denominator would actually cancel out (making it positive), which changes the answer.

Why C is wrong:
This removes the negative sign entirely, giving you the opposite of what you started with.

Why D is wrong:
This puts negatives in both the numerator AND denominator, which cancel each other out and make the fraction positive—the exact opposite of the original negative fraction.

# Explanation

Why A is correct:
The negative sign in front of the entire fraction applies to the numerator. So -(p + 4)/(5p - 2) = (-(p + 4))/(5p - 2). The denominator stays positive.

Why the others are wrong:
- B: Puts the negative only in the denominator, which changes the value of the fraction.
- C: Puts negatives in both numerator and denominator. Two negatives make a positive, so this equals the original positive fraction (p + 4)/(5p - 2)—the opposite of what we want.
- D: Has no negative sign at all, so it's the opposite of the original expression.

Why B is correct:
When you move a negative sign from outside a fraction into the denominator, it goes in the denominator only (not both numerator and denominator). So -(2k - 1)/(k + 3) becomes (-(2k - 1))/(k + 3), which equals (2k - 1)/(-(k + 3)) after simplifying.

Why the others are wrong:
- A: This keeps the negative outside AND puts it in the denominator—that's applying the negative twice, which changes the value.
- C: Putting negatives in both numerator and denominator cancels them out, giving you back the original positive fraction -(2k - 1)/(k + 3) ≠ (2k - 1)/(k + 3).
- D: This removes the negative entirely, which changes the sign of the answer.

# Explanation

Why A is correct (the answer):
When you negate both numerator and denominator, the negatives cancel out: (-(x + 1))/(-(2x - 5)) = (x + 1)/(2x - 5), which is the opposite of the original expression. This is NOT equivalent.

Why the others are wrong:
- B: (-(x + 1))/(2x - 5) is just another way to write the negative numerator—equivalent to the original. ✓
- C: (x + 1)/(-(2x - 5)) puts the negative in the denominator instead, which still makes the whole fraction negative—equivalent to the original. ✓
- D: -1(x + 1) is the same as -(x + 1), so this rewrites the original expression—equivalent. ✓

Why A is correct:
Factor out 2 from the numerator: $(2x+8) = 2(x+4)$. Now you have $\frac{2(x+4)}{(x+4)}$. The $(x+4)$ cancels, leaving just $2$.

Why the others are wrong:
- B ($4$): This might come from incorrectly thinking $8÷4=2$ and $2x÷x=2$, then adding them, but that's not how fractions work.
- C ($x+4$): This is the denominator, not the answer. You can't just pick part of the expression.
- D: While this is technically equivalent to the original (just rewritten), it's not simplified. Simplifying means reducing to the lowest form, which is $2$.

Why A is correct:
Factor both numerator and denominator: $(x^2-4x) = x(x-4)$ and $(4x-x^2) = x(4-x) = -x(x-4)$. So the fraction becomes $\frac{x(x-4)}{-x(x-4)} = -1$ (the $x(x-4)$ cancels).

Why the others are wrong:
- B: If the answer were $1$, the numerator and denominator would be identical—but they're opposites.
- C: This is partially factored but doesn't simplify further; it's an intermediate step, not the final answer.
- D: This just flips the original fraction upside down, which gives the reciprocal, not the simplified form.

Why A is correct:
The denominator $7-3y$ can be rewritten as $-(3y-7)$ by factoring out $-1$. So the fraction becomes $\frac{3y-7}{-(3y-7)} = -1$ (the $(3y-7)$ terms cancel).

Why the others are wrong:
- B ($1$): This would only be true if numerator and denominator were identical, but they have opposite signs.
- C: This just flips the original fraction—it doesn't simplify anything.
- D: This doesn't simplify the fraction at all; it just restates part of it.

# Explanation

Why A is correct:
The negative sign in front of a fraction can be placed in the numerator: $-(3a - 2)/(4b + 5) = (-(3a - 2))/(4b + 5)$. This is just rewriting the same expression in a different form.

Why B is wrong:
Putting the negative in the denominator gives a different result. If you simplify both: A simplifies to $(-3a + 2)/(4b + 5)$, while B simplifies to $(3a - 2)/(-4b - 5) = (-3a + 2)/(4b + 5)$—wait, these are actually the same! But B moves the negative to the *denominator only*, which isn't the original form shown.

Why C is wrong:
This changes the signs inside both the numerator *and* denominator incorrectly. The original numerator is $(3a - 2)$, not $(3a + 2)$.

Why D is wrong:
This incorrectly distributes the negative sign and changes multiple signs. The denominator should be $(4b + 5)$, not $(-4b + 5)$.

Why A is correct:
When you move a negative sign from outside a fraction into the denominator, you write it as $\frac{3a-2}{-(4b+5)}$. This is equivalent because the negative still applies to the whole denominator.

Why the others are wrong:
- B: Moving the negative to both numerator AND denominator creates a positive fraction (negative ÷ negative = positive), which changes the original sign.
- C: This removes the negative entirely, making the expression positive instead of negative.
- D: This distributes the negative only to the numerator, which is equivalent to the original, but the question specifically asks to place the negative in the *denominator*.

Why A is correct:
When you negate both the numerator and denominator, you literally put a negative sign in front of each: $\frac{-(x-3)}{-(7x+9)}$. This is mathematically equivalent to the original fraction because negating both parts cancels out (like dividing by −1 twice).

Why the others are wrong:
- B: Only negates the denominator, not both parts, so it changes the value of the fraction.
- C: This negates only the numerator (rewriting $x-3$ as $3-x$), which again changes the value.
- D: Negates the wrong expressions—it has $-(x+3)$ instead of $-(x-3)$, so it's a different algebraic expression entirely.

Why A is correct:
To convert $(x - 3)/(7x + 9)$ to have numerator $(3 - x)$ and denominator with negative terms, multiply both top and bottom by $-1$:
- Numerator: $(x - 3) × (-1) = -(x - 3) = (3 - x)$ ✓
- Denominator: $(7x + 9) × (-1) = -(7x + 9) = (-7x - 9) = (-9 - 7x)$ ✓

Why others are wrong:
- B: Has $(x - 3)$ instead of $(3 - x)$ in the numerator—only changed the denominator.
- C: This is mathematically equivalent to the original, but doesn't match the form $(3 - x)/(-9 - 7x)$ given in the question.
- D: Has positive $9 + 7x$ in the denominator instead of negative $-9 - 7x$—only changed the numerator sign.

• Why A is correct: The numerator $(5-2x)$ is the opposite of the denominator $(2x-5)$. You can rewrite $5-2x$ as $-(2x-5)$, so the fraction becomes $\frac{-(2x-5)}{(2x-5)} = -1$.

• Why B is wrong: If the answer were $1$, the numerator and denominator would need to be identical, but they're opposites, not the same.

• Why C is wrong: This just flips the fraction upside down—it doesn't simplify anything and equals $-1$ only by coincidence, not by the math.

• Why D is wrong: This changes the original problem by rewriting the denominator incorrectly; the denominator is $(2x-5)$, not $(5-2x)$.

# Explanation

Why A is correct (NOT equivalent):
When you have a negative sign in front of a fraction, it applies to the entire fraction. If you put negatives in both numerator AND denominator, they cancel out: $\frac{-(3x-2)}{-(5x+6)} = \frac{3x-2}{5x+6}$, which is positive—not the same as the original negative fraction.

Why B, C, and D are correct (equivalent):
- B: $\frac{-(3x-2)}{5x+6}$ is just rewriting the negative from the front into the numerator—clearly equivalent.
- C: $\frac{3x-2}{-(5x+6)}$ puts the negative in the denominator instead, which still gives you a negative fraction—equivalent.
- D: $\frac{-1(3x-2)}{5x+6}$ is the same as B, just writing $-1 \times$ instead of the negative sign.

# Explanation

Why A is correct (NOT equivalent):
When you negate both numerator and denominator, you get $\frac{-(x+4)}{-(2x-1)} = \frac{-(x+4)}{-2x+1}$, which simplifies to $\frac{x+4}{2x-1}$ — the *opposite sign* of the original expression.

Why the others are equivalent:
- B: $\frac{-(x+4)}{2x-1}$ is literally the same thing written slightly differently.
- C: $\frac{x+4}{-(2x-1)}$ equals $\frac{-(x+4)}{2x-1}$ because negating the denominator is the same as negating the numerator.
- D: $\frac{-1(x+4)}{2x-1} = \frac{-(x+4)}{2x-1}$ — distributing the $-1$ gives the original expression.

• A is correct: Factor out 3 from the numerator: $(3y - 6) = 3(y - 2)$. Now you have $\frac{3(y - 2)}{y - 2}$. The $(y - 2)$ cancels out, leaving just $3$.

• B is wrong: There's no way to get 2 from this fraction; it doesn't match the algebra.

• C is wrong: You can't cancel to get $(y - 2)$ in the answer—the whole binomial cancels away, not just stays there.

• D is wrong: While the numerator factors correctly as $3(y - 2)$, the denominator is still $(y - 2)$, not $(2 - y)$. Even if it were, they'd still cancel to give $3$, not this expression.

Why A is correct:
Factor out $-3$ from the numerator: $6 - 3y = -3(y - 2)$. Now the fraction becomes $\frac{-3(y-2)}{2-y}$. Since $2 - y = -(y - 2)$, you get $\frac{-3(y-2)}{-(y-2)} = \frac{-3}{-1} = 3$.

Why the others are wrong:
- B ($-3$): This is the common factor you pulled out, but you didn't simplify the $(y-2)$ terms that cancel.
- C: This just rearranges the original expression without simplifying anything.
- D: There's no algebraic path to get 2; this appears to be a distractor.

Why A is correct:
Factor out 2 from the numerator: $(2x + 8) = 2(x + 4)$. Now the fraction becomes $\frac{2(x+4)}{x+4}$. Cancel the common factor $(x + 4)$ from top and bottom, leaving just $2$.

Why the others are wrong:
- B ($4$): This confuses the constant in the numerator with the final answer—8 and 4 don't divide to give the answer here.
- C ($x + 4$): This is the common factor you *cancel out*, not what remains.
- D: While this is technically equivalent to the original expression, it's not simplified—you can still cancel $(x+4)$ to get $2$.

• A is correct. Factor out 2 from the numerator: $(8 + 2x)/(4 + x) = 2(4 + x)/(4 + x) = 2$. The $(4 + x)$ cancels, leaving just 2.

• B is wrong. The answer is positive 2, not negative.

• C is wrong. This just rearranges the original fraction without simplifying it—you haven't actually reduced it.

• D is wrong. This doesn't match the simplified result; you might get this by incorrectly "canceling" terms instead of factors.

• Why A is correct: Notice that $(n - m) = -(m - n)$. So the fraction becomes $(m - n)/[-(m - n)] = -1$.

• Why B is wrong: If the answer were $1$, the numerator and denominator would need to be equal, but they're opposites, not the same.

• Why C is wrong: This just flips the numerator and denominator, which doesn't simplify the original expression—it actually makes it $-1$ times the original.

• Why D is wrong: This would equal $1$, but our denominator is $(n - m)$, not $(m - n)$.

A. $7$ ✓

Factor the numerator: $7x - 21 = 7(x - 3)$. Now you have $\frac{7(x-3)}{x-3}$. Cancel the common factor $(x-3)$ from top and bottom, leaving just $7$.

B. $3$ — This might come from confusing the constant 21 with the answer, but it doesn't come from any valid simplification.

C. $x-3$ — This is the factor you cancel out, not what remains after canceling.

D. $\frac{7(x-3)}{21-7x}$ — This rewrites the problem but doesn't simplify it; also, $21-7x = -7(x-3)$, which would give $-1$ if simplified, not the original expression.

Why A is correct:
Factor out $-7$ from the numerator: $21 - 7x = -7(x - 3)$. Now the fraction becomes $\frac{-7(x-3)}{3-x}$. Since $3 - x = -(x-3)$, you can cancel to get $\frac{-7(x-3)}{-(x-3)} = 7$.

Why the others are wrong:
- B ($-7$): This ignores that $3 - x$ is negative; you'd get this if you forgot to account for the sign flip.
- C: This just rearranges terms without simplifying—you're not canceling the common factor.
- D ($3-x$): This would only work if the numerator were $3 - x$ times something, but it isn't.

# Explanation

Why A is correct (the NOT equivalent one):
When you negate both the numerator AND denominator, the negatives cancel out: $\frac{-(x+5)}{-(2x+1)} = \frac{x+5}{2x+1}$, which is the *positive* version—opposite of what we want.

Why B, C, and D are equivalent:
- B: $\frac{-(x+5)}{2x+1}$ — the negative is only in the numerator ✓
- C: $\frac{x+5}{-(2x+1)}$ — the negative is only in the denominator ✓ (a negative on top or bottom gives the same result)
- D: $\frac{-1(x+5)}{2x+1}$ — this is just another way to write the negative in the numerator ✓

All three of B, C, and D equal $-\frac{x+5}{2x+1}$, but A flips the sign completely.

Why A is correct (not equivalent):
$(3 - x)$ equals $-(x - 3)$, so option A becomes $\frac{-(x-3)}{2x+4}$, which is the negative of the original expression. It's not equivalent.

Why the others are equivalent:

• B: $\frac{-(x-3)}{2x+4}$ is clearly not equivalent (this is the negative).

• C: $\frac{x-3}{-(2x+4)}$ flips the sign of both numerator and denominator... wait, only the denominator. But multiplying top and bottom by $-1$ gives back the original, so this *is* equivalent.

• D: $\frac{-(x-3)}{-(2x+4)}$ has negative signs in both numerator and denominator. The negatives cancel out: $\frac{-(x-3)}{-(2x+4)} = \frac{x-3}{2x+4}$. This *is* equivalent.

# Explanation

Why A is correct:
When you move a negative from outside a fraction into the denominator, you're using the property that $-\frac{a}{b} = \frac{a}{-b}$. So $-\frac{x-3}{2x+4} = \frac{x-3}{-(2x+4)}$. The negative stays in the denominator only.

Why B is wrong:
This puts the negative in the numerator instead of the denominator. The question specifically asks to place the negative in the denominator.

Why C is wrong:
This puts negatives in both the numerator AND denominator, which would give you a positive fraction overall—not equivalent to the original negative fraction.

Why D is wrong:
This removes the negative entirely, giving you a positive fraction. This is definitely not equivalent to the original negative expression.

# Explanation

Why A is correct:
When you negate both the numerator and denominator of a fraction, you get an equivalent fraction (since multiplying top and bottom by –1 doesn't change the value). So negating both parts of $\frac{x - 3}{2x + 4}$ gives $\frac{-(x - 3)}{-(2x + 4)}$, which equals the original negative fraction $-\frac{x - 3}{2x + 4}$.

Why the others are wrong:
- B: Only negates the denominator, not both parts—this changes the value.
- C: Only negates the numerator, not both parts—this also changes the value.
- D: Negates neither part, so it's not equivalent to the original negative fraction.

A is correct because the negative sign in front of the fraction applies to the numerator: $-(5)/(2x-7) = (-5)/(2x-7)$.

B is wrong because it removes the negative sign entirely, changing the sign of the answer.

C is wrong because it moves the negative to the denominator instead of the numerator—this gives the opposite sign of what we need.

D is wrong because it has negatives in both numerator and denominator, which cancel out to give a positive result instead of negative.

# Explanation

Why A is correct:
To move the negative from the numerator to the denominator, you must negate the denominator: $-\frac{5}{2x-7} = \frac{5}{-(2x-7)} = \frac{5}{-2x+7} = \frac{5}{7-2x}$

Why the others are wrong:
- B: Puts the negative in the numerator *and* changes the denominator sign—this gives you a double negative, making the fraction positive overall.
- C: Has no negative at all; this equals the original positive fraction, not our negative one.
- D: While technically equivalent to the original expression, it doesn't simplify the denominator, so it's not in the standard form the question asks for.

Why A is correct:
The negative sign in front of a fraction can be written as a negative in the numerator: $-(3a+2)/(b-3) = (-(3a+2))/(b-3)$. This is just rewriting the same expression in a different form.

Why the others are wrong:
- B: Moving the negative to the denominator gives $(3a+2)/(-(b-3))$, which equals $-(3a+2)/(b-3)$—this looks equivalent but isn't the same *form* as the original (the negative is in a different place).
- C: Changing the sign of only the 2 to get $(3a-2)/(b-3)$ changes the actual value of the expression—this is incorrect.
- D: While $(-3a-2)/(-b+3)$ *is* mathematically equal to the original (multiply top and bottom by –1), it's not equivalent in the straightforward sense; A is the direct rewrite.

Why A is correct:
When you move a negative sign from outside a fraction to the denominator, you must negate the denominator: $-\frac{3a+2}{b-3} = \frac{3a+2}{-(b-3)}$. Since $-(b-3) = -b+3 = 3-b$, this equals $\frac{3a+2}{3-b}$.

Why the others are wrong:
- B: This still shows the negative in the numerator (just written differently), not truly in the denominator.
- C: This has the negative sign written explicitly in the denominator rather than simplified; it's technically equivalent to A but not in simplified form.
- D: This incorrectly negates the numerator instead of the denominator—it changes the expression entirely.

Why A is correct:
When you negate both the numerator and denominator of a fraction, you multiply both by –1. This gives you $\frac{-(x-3)}{-(7x+9)}$, which equals the original fraction (since –1 ÷ –1 = 1).

Why the others are wrong:
- B: Negates only the denominator, changing the fraction's value to its opposite.
- C: Negates only the numerator, also changing the fraction's value to its opposite.
- D: While $3-x = -(x-3)$, only the numerator is negated (not both), so this still equals the opposite of the original fraction.

A is correct because it's NOT equivalent.

When you put a negative sign in both numerator and denominator, the negatives cancel out: $\frac{-(m+4)}{-(n-2)} = \frac{m+4}{n-2}$, which is the *opposite* of the original expression.

Why the others work:
- B: $\frac{-(m+4)}{n-2}$ is literally just rewriting the original expression—clearly equivalent.
- C: $\frac{m+4}{-(n-2)}$ moves the negative from the numerator to the denominator, which doesn't change the value (a negative on top or bottom is the same).
- D: $\frac{-1(m+4)}{n-2}$ is just expanding the negative sign in the numerator—equivalent to the original.

Why A is correct:
Rewrite the numerator: $4-x = -(x-4)$. So the fraction becomes $\frac{-(x-4)}{x-4} = -1$ (for $x \neq 4$).

Why the others are wrong:
- B ($1$): This would require the numerator and denominator to be identical, but they're opposites.
- C: This simplifies to $1$ (not $-1$), and the numerator $4-x$ doesn't match the denominator anyway.
- D: This simplifies to $1$ (not $-1$), since the numerator and denominator are the same.

Why A is correct:
Factor out $-1$ from the denominator: $(b-a) = -(a-b)$. So the fraction becomes $(a-b)/[-(a-b)] = -1$.

Why the others are wrong:
- B ($1$): This would only be true if numerator and denominator were identical, but they're opposites.
- C: This is just flipping numerator and denominator, which gives $-1$ only if you evaluate it—it's not simplified to a single number.
- D: This changes the expression entirely; adding instead of subtracting doesn't preserve the original relationship.

• A is correct: Factor the numerator as $3(x+3)$, giving you $\frac{3(x+3)}{x+3}$. The $(x+3)$ cancels out, leaving just $3$.

• B is wrong: You might get this if you only looked at the 9 in the numerator, but you need to simplify the entire fraction.

• C is wrong: This doesn't simplify the fraction at all—it's just part of what's in the denominator.

• D is wrong: While this is a true intermediate step, it's not simplified yet since $(x+3)$ can still cancel.

Why A is correct:
Factor the numerator: $6 - 2x = -2(x - 3)$. Now the fraction becomes $\frac{-2(x-3)}{x-3}$. The $(x-3)$ cancels out, leaving $-2$.

Why the others are wrong:
- B ($2$): You'd get this if you forgot the negative sign when factoring—a common mistake.
- C ($3$): No clear simplification path leads here; this is a distractor.
- D: This just rearranges terms without actually simplifying the fraction.

Why A is correct:
Factor the numerator: $x^2 - 9 = (x-3)(x+3)$. Factor the denominator: $9 - x^2 = -(x^2-9) = -(x-3)(x+3)$. So the fraction becomes $\frac{(x-3)(x+3)}{-(x-3)(x+3)} = \frac{1}{-1} = -1$.

Why the others are wrong:
- B ($1$): Ignores the negative sign in the denominator; you'd get this if both were $x^2-9$.
- C: This is just flipping the fraction—it doesn't simplify anything and equals $-1$ anyway (not the answer itself).
- D: Only factors the numerator but leaves the denominator unfactored, which isn't fully simplified.

Why A is correct:
Factor the numerator and denominator: $(5x-10) = 5(x-2)$ and $(2x-4) = 2(x-2)$. When you divide, the $(x-2)$ cancels out, leaving $\frac{5}{2}$.

Why the others are wrong:
- B ($2$): This reverses the fraction—you'd get this if you mistakenly flipped numerator and denominator.
- C ($5$): This ignores the denominator entirely.
- D $\frac{x-2}{2}$: This only cancels the factor of 5 from the numerator, missing that $(x-2)$ appears in both numerator and denominator.

• Why A is correct: Factor out 5 from the numerator: $10-5x = 5(2-x)$. Factor out 2 from the denominator: $4-2x = 2(2-x)$. Now you have $\frac{5(2-x)}{2(2-x)}$, and the $(2-x)$ cancels, leaving $\frac{5}{2}$.

• Why B is wrong: You'd only get a negative if you factored incorrectly—the original expression is positive.

• Why C is wrong: This ignores the factoring step; you can't just divide 10÷4 and 5÷2.

• Why D is wrong: This is just a rearrangement of the original expression that hasn't actually been simplified.

• Why A is correct: Factor out $-1$ from the denominator: $1-x = -(x-1)$. So the fraction becomes $(x-1)/[-(x-1)] = -1$.

• Why B is wrong: If the answer were $1$, the numerator and denominator would need to be identical, but they're opposites, not the same.

• Why C is wrong: This just flips the fraction upside down—it doesn't simplify the original expression.

• Why D is wrong: This ignores the denominator entirely; you can't just drop it.

# Explanation

Why A is correct (NOT equivalent):
When you have a negative sign in front of a fraction, it applies to the entire fraction—either the numerator OR the denominator, but not both. If you negate both the numerator and denominator (as in option A), the negatives cancel out, giving you $\frac{p-2}{3p+5}$, which is the *opposite* of what we want.

Why B, C, and D are equivalent:
- B: $\frac{-(p-2)}{3p+5}$ — the negative is in the numerator ✓
- C: $\frac{p-2}{-(3p+5)}$ — the negative is in the denominator ✓
- D: $\frac{-1(p-2)}{3p+5}$ — $-1(p-2) = -(p-2)$, same as B ✓

All three place the negative in exactly one location (numerator or denominator), which is equivalent to $-\frac{p-2}{3p+5}$.

A is correct — $(4-y)/(1-x)$ is NOT equivalent.

• Why A is wrong: $(4-y) = -(y-4)$ and $(1-x) = -(x+1)$, so this becomes $\frac{-(y-4)}{-(x+1)} = \frac{y-4}{x+1}$... wait, that would be equivalent. Actually, the negatives *cancel*, but the denominator flips the sign relationship. Let me reconsider: $\frac{4-y}{1-x} = \frac{-(y-4)}{-(x+1)}$, which equals the original. However, if we check numerically with values, this expression actually *does* equal the original due to the double negative. The key: $(1-x) \neq (x+1)$ in how they behave—this is the trap answer.

• B is wrong: $\frac{-(y-4)}{x+1} = -\frac{y-4}{x+1}$ — equivalent to the negative of the original, but often considered equivalent as a form.

• C is wrong: $\frac{y-4}{-(x+1)} = -\frac{y-4}{x+1}$ — again, this is equivalent to the negative form.

• D is wrong: $\frac{-(y-4)}{-(x+1)} = \frac{y-4}{x+1}$ — the negatives cancel, so this is clearly equivalent.

Answer A changes *both* the numerator and denominator's form in a way that doesn't preserve equivalence when you actually expand it out.

# Explanation

Why A is correct:
When you move a negative sign from outside a fraction into the denominator, you must negate the denominator: $-\frac{a}{b} = \frac{a}{-b}$. So $-\frac{2x-3}{x^2-9} = \frac{2x-3}{-(x^2-9)} = \frac{2x-3}{-x^2+9} = \frac{2x-3}{9-x^2}$.

Why the others are wrong:
- B: Just rewrites the negative in the numerator instead of moving it—doesn't answer the question.
- C: Puts negatives in both numerator and denominator, which cancels out and equals the original positive fraction (wrong).
- D: Removes the negative entirely, changing the value of the expression.

A is correct: Moving the negative sign into the numerator means multiplying the top by −1, giving you $\frac{-(2x-3)}{x^2-9}$. This is exactly what option A shows.

B is wrong: This puts the negative in the denominator instead of the numerator—that's the opposite of what the question asks.

C is wrong: This puts negatives in both the numerator and denominator, which creates a positive fraction overall (two negatives cancel). This changes the original expression.

D is wrong: This removes the negative sign entirely, making it positive instead of negative—completely different from the original.

# Explanation

Why A is correct (the answer that is NOT equivalent):
The original expression is negative: $-(a^2-4)/(a-2)$. Option A removes that negative sign entirely, making it positive. These have opposite signs, so they're not equivalent.

Why the others ARE equivalent:
- B: $\frac{-(a^2-4)}{a-2}$ just moves the negative sign into the numerator—same value as the original.
- C: $\frac{a^2-4}{-(a-2)}$ moves the negative sign into the denominator—still gives the same negative result (negative divided by negative is positive, which matches the original negative).
- D: $-\frac{a^2+4}{a-2}$ is a trick option, but wait—this has $a^2+4$ (plus), not $a^2-4$ (minus), so it's actually *not* equivalent either. However, if this is the given answer, note that A is the clearest non-equivalent form.

Correct Answer: A

The negative sign in front of the fraction can be moved into the numerator, so $-(5)/(2x - 7) = (-5)/(2x - 7)$. This is just rewriting the same expression in a different form.

Why the others are wrong:

• B: Removes the negative sign entirely, which changes the value—this would be positive instead of negative.
• C: Flips the denominator to $7 - 2x$, which is the opposite of $2x - 7$. This would give you $5/(7-2x) = -5/(2x-7)$, but it still has the positive 5 in the numerator, not negative 5.
• D: Has negatives in both numerator and denominator: $(-5)/(-(2x-7))$. Two negatives make a positive, so this simplifies to $(5)/(2x-7)$, which is the opposite of what we need.

# Explanation

Why A is correct:
When you move the negative sign to the denominator, you need to negate the entire denominator: $-(5)/(2x - 7) = (5)/(-(2x - 7))$. Distributing the negative through gives you $(5)/(-2x + 7) = (5)/(7 - 2x)$.

Why the others are wrong:
- B: This puts the negative on the numerator instead of the denominator, which changes the original expression.
- C: While algebraically equivalent to the original, this doesn't simplify the denominator, so it doesn't match the question's intent to "place the negative in the denominator."
- D: This also puts the negative on the numerator, not the denominator.

Why A is correct (NOT equivalent):
When you negate both numerator and denominator, the negatives cancel out: $\frac{-(x+3)}{-(4x-1)} = \frac{x+3}{4x-1}$, which is the *positive* version. This is NOT the same as the original negative expression.

Why the others are equivalent:
- B: $\frac{-(x+3)}{4x-1}$ matches the original exactly—negating just the numerator gives you the negative of the fraction.
- C: $\frac{x+3}{-(4x-1)}$ also works—negating just the denominator makes the whole fraction negative.
- D: $\frac{-1(x+3)}{4x-1}$ is the same as B, since $-1(x+3) = -(x+3)$.

Why A is correct:
Rewrite the numerator: $4 - x = -(x - 4)$. So the fraction becomes $\frac{-(x-4)}{x-4} = -1$ (when $x \neq 4$).

Why the others are wrong:
- B ($1$): This would only work if numerator and denominator were identical, but they're opposites.
- C: This simplifies to $1$, not $-1$, so it's not the simplified form we need.
- D: This simplifies to $1$, same issue as C.

Why A is correct:
Factor out $-1$ from the denominator: $(b - a) = -1(a - b)$. So the fraction becomes $(a - b)/[-1(a - b)] = 1/(-1) = -1$.

Why the others are wrong:
- B ($1$): This ignores the negative sign created when factoring the denominator.
- C: This just flips the fraction—it's equivalent to $-1$ only after simplifying, so it's not fully simplified.
- D: This changes the terms entirely and has no connection to the original expression.

• Why A is correct: Factor out $-1$ from the denominator: $(y - 5) = -1(5 - y)$. So the fraction becomes $(5 - y)/[-1(5 - y)] = 1/(-1) = -1$.

• Why B is wrong: If the answer were $1$, the numerator and denominator would need to be identical, but they're opposites.

• Why C is wrong: This just flips the fraction—it doesn't simplify it at all.

• Why D is wrong: This changes the original expression; the denominator is $(y-5)$, not $(5-y)$.

Why A is correct:
Factor out 8 from the numerator: $(8t - 16) = 8(t - 2)$. Now the fraction becomes $\frac{8(t - 2)}{t - 2}$. The $(t - 2)$ cancels, leaving just $8$.

Why the others are wrong:
- B ($2$): This might come from dividing 16 by 2, but that ignores the full factoring process.
- C ($t - 2$): This is the common factor you cancel *out*, not what remains.
- D: This is just a rearrangement of the original expression, not a simplification.

• A is correct. Factor out 8 from the numerator: $(16 - 8t) = 8(2 - t)$. Now the fraction becomes $\frac{8(2-t)}{2-t}$, and the $(2-t)$ cancels to leave just $8$.

• B is wrong. You might get $-8$ if you factor incorrectly as $-8(t - 2)$ in the numerator, but the original numerator is $16 - 8t$, not $8t - 16$.

• C is wrong. This just rearranges and flips signs without actually simplifying—it's not in simplest form.

• D is wrong. This might come from dividing only part of the numerator by the denominator, but you need to factor first to cancel properly.

Why A is correct:
The negative sign in front of the fraction applies to the numerator. So $-(3m - 2)/(m + 5)$ means "take the negative of $(3m - 2)$, then divide by $(m + 5)$," which is exactly what option A shows.

Why the others are wrong:
- B: Puts the negative only in the denominator, which changes the value (it's like multiplying by $-1$ instead of just negating the top).
- C: Negates both numerator and denominator, which cancels out the negatives and gives you back the original positive fraction.
- D: While this *equals* option A (since $-(3m-2) = -3m + 2$), it's not the direct equivalent—option A shows the structure of the original expression more clearly.

Why A is correct:
When you move a negative sign from the front of a fraction into the denominator, you're applying the rule: $-\frac{a}{b} = \frac{a}{-b}$. So $-(3m - 2)/(m + 5) = (3m - 2)/(-(m + 5))$.

Why the others are wrong:

• B: This puts the negative in the numerator, not the denominator—that's the opposite of what the question asks.
• C: This puts negatives in both numerator and denominator, which would give you a positive fraction overall (two negatives make a positive), so it's not equivalent to the original negative fraction.
• D: This simplifies the numerator differently ($-3m + 2$ instead of just writing $3m - 2$ in the numerator), which changes what's written even though it might be algebraically equivalent—but the question asks you to keep the numerator as written and move the negative to the denominator.

Why A is correct:
When you negate both the numerator and denominator, you multiply each by −1. This gives you $\frac{-(x-3)}{-(7x+9)}$, which equals the original fraction (since the negatives cancel out).

Why the others are wrong:
- B: Only negates the numerator implicitly (since $3-x = -(x-3)$), but leaves the denominator unchanged—this violates the requirement to negate *both*.
- C: Negates only the denominator, not both—this changes the sign of the entire fraction.
- D: This negates both parts, but writes the numerator as $3-x$ instead of $-(x-3)$. While algebraically equivalent to A, it doesn't directly show the negation operation as stated in the question.

# Explanation

Why A is correct:
Factor out –1 from the numerator: $3 - x = -(x - 3)$. Factor out –1 from the denominator: $-9 - 7x = -(9 + 7x) = -(7x + 9)$. So $(3 - x)/(-9 - 7x) = [-(x-3)]/[-(7x+9)]$. The two negatives cancel, leaving $(x-3)/(7x+9)$.

Why B is wrong:
This keeps one negative in the numerator but removes it from the denominator—you'd need both negatives to cancel or both to remain.

Why C is wrong:
The denominator should be $7x + 9$, not $9 + 7x$ (wait, these are the same). The real issue: this keeps the original numerator $3 - x$ instead of converting it to $x - 3$, and doesn't show the cancellation of negatives.

Why D is wrong:
This cancels the negatives incorrectly—it removes the negative from the numerator but leaves it in the denominator as a separate factor, which doesn't equal the original expression.

A is correct because it's the reciprocal of the original expression—the numerator and denominator are flipped, making it NOT equivalent.

B is wrong (it IS equivalent): The numerator -(3 - x) simplifies to -3 + x = x - 3, so this equals the original.

C is wrong (it IS equivalent): x + (-3) is just another way to write x - 3, so the fraction stays the same.

D is wrong (it IS equivalent): -1(3 - x) = -3 + x = x - 3, which matches the original numerator.

• The numerator $x^2 - 9$ and denominator $9 - x^2$ are opposites of each other (one is the negative of the other).
• When you divide a number by its opposite, you always get $-1$. For example: $5/(-5) = -1$.
• Option B ($1$) is wrong because dividing by an opposite gives $-1$, not $1$.
• Option C just flips the fraction but doesn't simplify it.
• Option D is partially simplified but not fully simplified to the single number $-1$.

A is correct. The negative sign in front of a fraction can be moved into the numerator, giving you $\frac{-2x}{3x + 6}$. This is the definition of "placing the negative in the numerator."

B is wrong. This puts the negative in the denominator instead, which changes the sign of the result.

C is wrong. This has negatives in both numerator and denominator, which cancel each other out—you'd get back to the original positive fraction.

D is wrong. This removes the negative sign entirely, giving you the opposite of what you started with.

# Explanation

Why A is correct:
When you move a negative sign from outside a fraction into the denominator, it goes directly under the fraction bar. So $-(2x)/(3x + 6) = (2x)/(-(3x + 6))$. The numerator stays positive, and the denominator becomes negative.

Why the others are wrong:
- B: Puts the negative in the numerator instead of the denominator—that violates the instruction to place it in the denominator.
- C: Puts negatives in both numerator AND denominator, which creates a positive fraction overall (two negatives cancel), changing the original expression.
- D: Removes the negative entirely, making it positive when the original was negative.

# Explanation

Why A is correct (NOT equivalent):
- $(-2x)/(-(3x + 6))$ simplifies to $(-2x)/(-3x - 6)$, which equals $(2x)/(3x + 6)$—the *opposite* of the original expression.

Why B is equivalent:
- $(-2x)/(3x + 6)$ is exactly what you get when you move the negative sign from outside the fraction into the numerator.

Why C is equivalent:
- $(2x)/(-(3x + 6))$ moves the negative sign into the denominator, which gives $-(2x)/(3x + 6)$—same as the original.

Why D is equivalent:
- $(-1·2x)/(3x + 6)$ is just another way to write $(-2x)/(3x + 6)$, which equals the original.