Fill in the correct inequality sign between 3 - 2a and 3 - 2b, given the relationship between a and b.

Step-by-step solution

1. 1

   Work backwards from the two expressions. To compare $3-2a$ and $3-2b$, undo the operations that built them from $a$ and $b$: first the subtraction from $3$, then the multiplication by $-2$. Each step must be checked for whether it preserves or reverses order.

2. 2

   Subtracting the same constant preserves order. Removing $3$ from both sides leaves the comparison

   $-2a \quad \underline{\phantom{<}} \quad -2b$

   Adding or subtracting the same number never changes an inequality's direction.

3. 3

   Dividing by −2 reverses order. This is the only rule that flips the sign. So the relationship between $-2a$ and $-2b$ is the opposite of the relationship between $a$ and $b$, and the same holds after adding $3$ back.

4. 4

   State the three cases.

   $a < b \;\Longrightarrow\; 3-2a > 3-2b, \qquad a > b \;\Longrightarrow\; 3-2a < 3-2b, \qquad a = b \;\Longrightarrow\; 3-2a = 3-2b$

5. 5

   Verify with numbers. Take $a = 1$, $b = 4$ (so $a < b$): $3 - 2 = 1$ and $3 - 8 = -5$, and indeed $1 > -5$. Take $a = 5$, $b = 2$: $3 - 10 = -7$ and $3 - 4 = -1$, and $-7 < -1$. The direction is reversed in both cases, exactly as predicted.

6. 6

   Compare with a positive coefficient. The expression $4a - 3$ versus $4b - 3$ behaves the opposite way: multiplying by the positive $4$ preserves order, so $a < b$ gives $4a - 3 < 4b - 3$. The sign of the multiplier is the whole story.