Question

Solve the following inequation and represent the solution set on the number line, $-3<-\frac{1}{2}-\frac{2x}{3}\le \frac{5}{6},\forall x\in \mathrm{ℝ}$.

Answer 1

Step 1: Given inequality: $-3<-\frac{1}{2}-\frac{2x}{3}\le \frac{5}{6}$.

Add all sides of the inequality by $\frac{1}{2}$.

$$\begin{gathered} -3+\frac{1}{2}<-\frac{1}{2}-\frac{2x}{3}+\frac{1}{2}\le \frac{5}{6}+\frac{1}{2} \\ \frac{-6+1}{2}<-\frac{2x}{3}\le \frac{5+3}{6} \\ -\frac{5}{2}<-\frac{2x}{3}\le \frac{4}{3} \end{gathered}$$

Step 2: Multiply all sides of the inequality by $\left(-\frac{3}{2}\right)$ and change the sign of the inequality.

$$\begin{gathered} -\frac{5}{2}\times \left(-\frac{3}{2}\right)>-\frac{2x}{3}\times \left(-\frac{3}{2}\right)\ge \frac{4}{3}\times \left(-\frac{3}{2}\right) \\ \frac{15}{4}>x\ge -2 \\ -2\le x<3\frac{3}{4} \end{gathered}$$

Therefore, the solution set of the given inequality can be given by, $[-2,3\frac{3}{4})$.

Step 3: Represent the solution set on the number line

The solution set of the given inequality can be plotted as,