Solve the inequality (2x - 1)/3 > (3x - 5)/4 and write the solution set.

Step-by-step solution

1. 1

   Clear the fractions with the least common denominator. The denominators are $3$ and $4$, so the LCD is $12$. Multiplying both sides of an inequality by a positive number keeps the direction of the inequality, and $12>0$, so nothing flips:

   $12\cdot\frac{2x-1}{3}>12\cdot\frac{3x-5}{4}\;\Longrightarrow\;4(2x-1)>3(3x-5).$

   This single move is why fraction inequalities are no harder than integer ones — get rid of the denominators first, before touching the $x$ terms.

2. 2

   Distribute on both sides. Expanding each product:

   $4(2x-1)=8x-4,\qquad 3(3x-5)=9x-15,$

   so the inequality becomes

   $8x-4>9x-15.$

   Watch the $-3\times(-5)=+15$ sign: dropping that minus is the most common error in this step, and it changes the final boundary from $11$ to $-19$.

3. 3

   Collect the $x$ terms on the side that keeps the coefficient positive. Subtracting $8x$ from both sides leaves the $x$ on the right:

   $-4>x-15.$

   You could instead subtract $9x$ and get $-x-4>-15$, but then you would have to divide by $-1$ and remember to reverse the inequality. Moving the smaller coefficient avoids that trap entirely.

4. 4

   Isolate $x$ and read off the solution. Adding $15$ to both sides:

   $11>x,\qquad\text{that is}\qquad x<11.$

   In interval notation the solution set is $(-\infty,11)$. Note $11$ itself is excluded because the original inequality is strict.

5. 5

   Test one value inside and one outside. At $x=10.9$: left side $=\frac{20.8}{3}\approx 6.933$, right side $=\frac{27.7}{4}=6.925$, and $6.933>6.925$ ✓. At $x=11.1$: left $\approx 7.067$, right $=7.075$, so the inequality fails ✓. The boundary $x=11$ makes both sides equal $7$, confirming it is exactly the cut-off.