Solve the inequality (2x - 5)/4 < (3x - 3)/3.

Step-by-step solution

1. 1

   Multiply by the least common multiple of the denominators. $\operatorname{lcm}(4,3) = 12$. Because $12 > 0$, the direction of the inequality is preserved — this is the one place an inequality differs from an equation, and it only becomes an issue when the multiplier is negative.

   $12 \cdot \frac{2x-5}{4} < 12 \cdot \frac{3x-3}{3} \quad\Longrightarrow\quad 3(2x-5) < 4(3x-3)$

2. 2

   Expand both sides.

   $6x - 15 < 12x - 12$

3. 3

   Collect the x terms on the side with the larger coefficient. Subtracting $6x$ keeps the coefficient positive, so no sign flip is needed later:

   $-15 < 6x - 12$

4. 4

   Isolate x.

   $-15 + 12 < 6x \quad\Longrightarrow\quad -3 < 6x \quad\Longrightarrow\quad x > -\frac{1}{2}$

5. 5

   Test the boundary and one point on each side. At $x = -\tfrac12$: left $= \tfrac{-6}{4} = -1.5$, right $= \tfrac{-4.5}{3} = -1.5$ — equal, so the endpoint is correctly excluded. At $x = 0$: $-1.25 < -1$, true. At $x = -1$: $-1.75 < -2$, false. The solution set is $\left(-\tfrac12,\ \infty\right)$.