Question

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

Answer 1

Step 1 (Solve the inequation):

$$\begin{gathered} -\frac{x}{3}\le \frac{x}{2}-1\frac{1}{3}<\frac{1}{6},x\in R \\ -\frac{x}{3}\le \frac{x}{2}-\frac{4}{3}<\frac{1}{6} \end{gathered}$$

multiply both sides by 6

$$\begin{gathered} -2x\le 3x-8<1 \\ -2x\le 3x-8and3x-8<1 \\ 8\le 5xand3x<9 \\ x\ge \frac{8}{5}andx<3 \end{gathered}$$

Step 2 (Representation on the number line):

The solution set will be$\frac{8}{5}\le x<3$, so the solution set on the number line will be

Hence the solution set of the inequality $-\frac{x}{3}\le \frac{x}{2}-1\frac{1}{3}<\frac{1}{6},x\in R$ is $\frac{8}{5}\le x<3$ or $[\frac{8}{5},3)$ and the representation of the solution set on the number line is as shown above.