Solve the inequality (7x − 1)/(3x − 2) < 4, explaining the reason for each step.

Step-by-step solution

1. 1

   Do not multiply both sides by 3x − 2. The quantity $3x-2$ is positive for $x>\tfrac23$ and negative for $x<\tfrac23$. Multiplying an inequality by an expression of unknown sign is invalid — it would preserve the direction on one part of the line and reverse it on the other. The safe move is to bring everything to one side and compare with zero.

2. 2

   Subtract 4 from both sides. Adding or subtracting a constant is always legitimate:

   $\frac{7x-1}{3x-2}-4<0$

3. 3

   Combine into a single fraction. Write $4$ as $\dfrac{4(3x-2)}{3x-2}$:

   $\frac{7x-1-4(3x-2)}{3x-2}=\frac{7x-1-12x+8}{3x-2}=\frac{7-5x}{3x-2}<0$

4. 4

   Locate the critical points. The numerator vanishes at $7-5x=0$, i.e. $x=\tfrac75$; the denominator vanishes at $x=\tfrac23$. These split the line into $\left(-\infty,\tfrac23\right)$, $\left(\tfrac23,\tfrac75\right)$ and $\left(\tfrac75,\infty\right)$.

5. 5

   Test one point in each interval.

   $x=0$: $\dfrac{7}{-2}<0$ ✓

   $x=1$: $\dfrac{2}{1}=2>0$ ✗

   $x=2$: $\dfrac{-3}{4}<0$ ✓

6. 6

   Assemble the solution, deciding each endpoint. The inequality is strict, so $x=\tfrac75$ (where the fraction is $0$) is excluded, and $x=\tfrac23$ is excluded because the expression is undefined there:

   $\boxed{x<\dfrac23\quad\text{or}\quad x>\dfrac75}$

7. 7

   Confirm against the original inequality. At $x=0$: $\tfrac{-1}{-2}=0.5<4$ ✓. At $x=1$: $\tfrac{6}{1}=6\not<4$ ✗. At $x=2$: $\tfrac{13}{4}=3.25<4$ ✓. Note that naive cross-multiplication would give $7x-1<12x-8$, i.e. $x>\tfrac75$ — correct on one branch but missing the entire interval $x<\tfrac23$, exactly as warned in step 1.