Question

Solve the linear Inequality. Express the solution using interval notation and graph the solution set.

$\frac{2}{3}x-\frac{1}{2}x\ge \frac{1}{6}+x$

Short answer

The solution set of the given inequality is $x\le -\frac{1}{5}$that is $x\in (-\infty ,-\frac{1}{5}]$.

Step-by-step solution

Step-1. Determining the solution to Inequality.

$\frac{2}{3}x-\frac{1}{2}x\ge \frac{1}{6}+x$

$\frac{2}{3}x-\frac{1}{2}x-x\ge \frac{1}{6}+x-x$ (Subtract x)

$-\frac{5}{6}x\ge \frac{1}{6}$ (Simplify)

$-\frac{6}{5}\times (-\frac{5}{6}x)\le \frac{1}{6}\times (-\frac{6}{5})$ (Multiply by $-\frac{6}{5}$and reverse the inequality)

$x\le -\frac{1}{5}$

Step-2. Expressing the solution set in Interval Notation.

The solution set consists of all numbers less than or equals to $-\frac{1}{5}$. In other words the solution of the inequality is in the interval$(-\infty ,-\frac{1}{5}]$

Step-3. Graphing the solution set.