Question

Solve the following inequalities and give the solutions in both set and interval notations.

Answer 1

Given:

The inequality is

$-\dfrac{x}{8}+\dfrac{8}{3}\geq 3$

To find:

The solution for the given inequality in both set and interval notations.

Solution:

We have,

$-\dfrac{x}{8}+\dfrac{8}{3}\geq 3$

It can be written as

$-\dfrac{x}{8}\geq 3-\dfrac{8}{3}$

$-\dfrac{x}{8}\geq \dfrac{9-8}{3}$

$-\dfrac{x}{8}\geq \dfrac{1}{3}$

Multiply both sides by -8 and change the inequality sign.

$x\leq -\dfrac{8}{3}$

It means all the values of $x$ which are less than or equal to $-\dfrac{8}{3}$ are included in the solution set.

Set notation is $\{x|x\leq -\dfrac{8}{3}\}$.

Interval notation is  $\left(-\infty,-\dfrac{8}{3}\right ]$.