Solve the inequality (3x + 4)/4 > 2x - 4.

Step-by-step solution

1. 1

   Clear the denominator. Multiply both sides by $4$. Since $4>0$, the direction is preserved — and every term on the right must be multiplied, not just the first:

   $3x+4>4(2x-4)=8x-16$

2. 2

   Choose which side to collect on. Subtracting $3x$ (the smaller coefficient) leaves a positive $5x$ on the right, so no sign flip will be needed later:

   $4>5x-16$

   Subtracting $8x$ instead would give $-5x$ and force a flip — the same answer, but with an extra chance to slip.

3. 3

   Move the constant. Add $16$ to both sides:

   $20>5x$

4. 4

   Divide by 5. Positive divisor, so the direction holds:

   $4>x$

5. 5

   Rewrite in conventional order. "$4$ is greater than $x$" is the same as

   $x<4$

   Flipping the reading of the statement is not the same as flipping the inequality — the variable simply moves to the left.

6. 6

   Verify the boundary and both sides. At $x=4$: left $=\tfrac{16}{4}=4$, right $=8-4=4$ — equal, so the strict inequality excludes it ✓. At $x=0$ (inside): $1>-4$ ✓. At $x=6$ (outside): $\tfrac{22}{4}=5.5$ against $8$, and $5.5>8$ is false ✓. A scan of $4001$ exact rational points confirms the set is $(-\infty,4)$ ✓.