Question

Solve each inequality. Describe the solution set using set-builder or interval notation. Then graph the solution set on number line.

$11-c\le 8$

Short answer

Solution of inequality $11-c\le 8$ is $3\ge c$

In interval notation solution set is $\left[3,\infty \right)$

Solution set on number line is

Step-by-step solution

- Write properties of inequality

Properties of inequalities are written in following table:

- Simplify 11−c≤8

To simplify given inequality, use addition and subtraction property of inequalities. Add or subtract same number from both sides does not change inequality as shown below

$\begin{array}{c}11-c\le 8\\ 11-c-8+c\le 8-8+c\\ 11-8-c+c\le c\\ 3\le c\end{array}$

- Write solution set

The solution set of the inequality $3\le c$ contains 3 less than or equal to all real numbers

In interval notation, solution set contains all real number between 3 to $\infty$

As $\infty$ is not an end points so parentheses is used with it whereas 3 is included in solution set so bracket is used with it.

Solution set is $\left[3,\infty \right)$

- Graph the solution set

Assolution set of the inequality $3\le c$ contains 3 less than or equal to all real number

So arrow is towards $\infty$ and dot on 3 denotes it is included in the solution set.