Q.Solve the compound inequality $-5 \le \dfrac{3x+2}{4} < 4$ for real $x$, and write the solution as an interval.

[!TLDR] Operate on all three parts of the compound inequality simultaneously: multiply by 4, then subtract 2, then divide by 3. [!AN …

Start with $-5 \le \dfrac{3x+2}{4} < 4$. Multiply all three parts by $4$, a positive number (no flip): $-20 \le 3x+2 < 16$. Subtract $2$ from all three parts: $-22 \le 3x < 14$. Divide all three parts by the positive number $3$: $-\dfrac{22}{3} \le x < \dfrac{14}{3}$. Check the left boundary $x=-22/3$: $3x+2 = -22+2=-20$, and $-20/4=-5$, which equals $-5$ — correctly included since the left inequality is $\le$. Check the right boundary $x=14/3$ …

Multiply, then subtract, then divide — applying every operation identically to all three pa …

A frequent mistake is solving the left and right halves separately with different, inconsistent operations, which can sil …