Let's walk through the problems in your image:

Part 1: Simplify if possible. Assume there are no zero denominators.

Problem (a) from the top part:

Expression: $\frac{2x - 6}{3 - x}$ You entered $-2$, which was marked correct. Let's quickly confirm:

• Factor the numerator: $2x - 6 = 2(x - 3)$.
• So, the expression becomes: $\frac{2(x - 3)}{3 - x}.$ Notice that $3 - x$ can be rewritten as $-(x - 3)$, so the expression simplifies to: $\frac{2(x - 3)}{-(x - 3)} = -2.$ This is indeed correct.

Problem (b) from the top part:

Expression: $\frac{3x - 1}{4 - 12x}.$ You simplified it as: $\frac{3x - 1}{4(1 - 3x)}.$ This is incorrect. Let's try simplifying:

• Factor the denominator: $4 - 12x = 4(1 - 3x)$.
• The expression becomes: $\frac{3x - 1}{4(1 - 3x)}.$ However, $3x - 1$ does not cancel with $1 - 3x$. The correct final expression remains: $\frac{3x - 1}{4(1 - 3x)}.$ Since it cannot be simplified further, the correct answer is just the original expression: $\frac{3x - 1}{4 - 12x}.$

Part 2: Simplify if possible.

Problem (a) from the second part:

Expression: $\frac{4 - 12x}{3x - 1}.$ This cannot be simplified any further. You entered the expression itself, which is correct. No changes are needed.

Problem (b) from the second part:

Expression: $\frac{-x - 3}{6 - 2x}.$ You entered: $-\frac{1}{2}.$ Let's check:

• Factor the denominator: $6 - 2x = 2(3 - x)$.
• The numerator can be written as $-(x + 3)$, which is $-(3 - x)$. Now the expression becomes: $\frac{-(3 - x)}{2(3 - x)}.$ The terms cancel, and you're left with: $-\frac{1}{2}.$ This is correct.

Summary:

• For Part 1(b): The correct answer should be the original expression, not the simplified version you entered. The answer is: $\frac{3x - 1}{4 - 12x}.$

Would you like further details or clarifications on any of these steps?

Here are 5 related questions to consider:

1. How can recognizing factoring help simplify rational expressions?
2. What is the significance of checking for denominator restrictions (i.e., values that make the denominator zero)?
3. Why do some expressions appear unsimplifiable even after factoring?
4. How can rewriting terms (like $3 - x$) help in simplifying expressions?
5. How do you determine when to leave an expression in its original form?

Tip: When simplifying, always factor both the numerator and denominator completely before attempting to cancel terms.