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

Step-by-step solution

1. 1

   Clear the fractions with the least common multiple. The denominators are $4$ and $3$, so their LCM is $12$. Multiplying an inequality by a positive number preserves the direction, so it is safe:

   $12\left(2x-\frac54\right)<12\left(3-\frac{x}{3}\right)$

2. 2

   Distribute on both sides. Every term must be multiplied, including the ones without fractions:

   $24x-15<36-4x$

   Check each: $12\cdot2x=24x$, $12\cdot\tfrac54=15$, $12\cdot3=36$, $12\cdot\tfrac{x}{3}=4x$ ✓.

3. 3

   Gather the x terms on the left. Add $4x$ to both sides — choosing to add rather than subtract keeps the coefficient positive and avoids a later sign flip:

   $28x-15<36$

4. 4

   Gather the constants on the right. Add $15$:

   $28x<51$

5. 5

   Divide by 28. Since $28>0$, the direction is unchanged:

   $x<\frac{51}{28}$

   The fraction does not reduce: $51=3\times17$ and $28=2^2\times7$ share no factor. As a decimal, $\tfrac{51}{28}\approx1.8214$.

6. 6

   Verify the boundary and one point on each side. At $x=\tfrac{51}{28}$ the two sides are exactly equal ✓. At $x=0$ (inside): $-\tfrac54=-1.25<3$ ✓. At $x=2$ (outside): $4-1.25=2.75$ versus $3-\tfrac23=2.333$, and $2.75<2.333$ is false ✓. A scan of $1000$ exact rational points confirms the solution set is precisely $\left(-\infty,\tfrac{51}{28}\right)$ ✓.