Q.Solve $|4 - 3x| \ge 5$ for real $x$; express the solution as a union of intervals, being careful with the sign when isolating $x$.

[!TLDR] Split into two cases and divide by the negative coefficient $-3$ in each, flipping both times. [!ANSWER] $x \in \ …

$|4-3x|\ge5$ splits into two cases: $4-3x\ge5$ or $4-3x\le-5$. Case 1: $4-3x\ge5 \Rightarrow -3x\ge1$; dividing by $-3$ (negative) reverses the inequality: $x\le-\dfrac{1}{3}$. Case 2: $4-3x\le-5 \Rightarrow -3x\le-9$; dividing by $-3$ (negative) reverses the inequality: $x\ge3$. So the solution is $x\le-\dfrac{1}{3}$ or $x\ge3$. Check $x=-1$: $|4-3(-1)|=|4+3|=7\ge5$ ✓; check $x=4$: $| …

Split into the two cases from the slack "greater than" modulus rule, then in each case isolate $x$ by dividing by the negative coeffici …

Because the coefficient of $x$ inside the modulus is negative ($-3x$), a student who forgets to flip the inequality when dividing by $-3$ in either case will …