Fill in the correct inequality sign between 4a - 3 and 4b - 3, given the relationship between a and b.

Step-by-step solution

1. 1

   Identify the two operations applied to a and b. Each expression is built by multiplying by $4$ and then subtracting $3$. Both operations need to be checked against the order rules.

2. 2

   Multiplying by a positive number preserves order. Since $4 > 0$,

   $a < b \;\Longrightarrow\; 4a < 4b$

   Only a negative multiplier would reverse the sign here.

3. 3

   Subtracting a constant preserves order. Removing $3$ from both sides shifts both values equally:

   $4a < 4b \;\Longrightarrow\; 4a - 3 < 4b - 3$

4. 4

   State the three cases.

   $a < b \;\Longrightarrow\; 4a-3 < 4b-3, \qquad a > b \;\Longrightarrow\; 4a-3 > 4b-3, \qquad a = b \;\Longrightarrow\; 4a-3 = 4b-3$

   The direction always matches the original — the map $t \mapsto 4t - 3$ is strictly increasing.

5. 5

   Verify with numbers. With $a = 1$, $b = 4$: $4 - 3 = 1$ and $16 - 3 = 13$, and $1 < 13$. With $a = -2$, $b = -5$ (so $a > b$): $-11 > -23$. Both preserve the original direction.

6. 6

   Contrast with a negative coefficient. For $3 - 2a$ versus $3 - 2b$ the multiplier is $-2$, so the order reverses: $a < b$ would give $3-2a > 3-2b$. Comparing the two problems isolates exactly which step causes a flip.