Q43. - Algebra 1

Question

Solve each inequality. Then graph the solution set.

$\left|m+19\right|\mathbf{\le }\mathbf{1}$

Step-by-Step Solution

Answer

The solution for the given inequality $\left|m+19\right|\le 1$ is $\mathit{m}\in \left[-\mathbf{20},-\mathbf{18}\right]$.

The graph of the solution set which is $m\in \left[-20,-18\right]$ is:

Step 1. Solve the given inequality | m + 19 | ≤ 1 .

The solution of the given inequality $\left|m+19\right|\le 1$ is:

Case 1: $m+19$ is non-negative.

$\begin{array}{c}m+19\le 1\\ m+19-19\le 1-19\\ m\le -18\\ m\in \left(-\infty ,-18\right]\end{array}$

Case 2: $m+19$ is negative.

$\begin{array}{c}-\left(m+19\right)\le 1\\ \left(-1\right)\left(-\left(m+19\right)\right)\ge \left(-1\right)\left(1\right)\\ m+19\ge -1\\ m+19-19\ge -1-19\\ m\ge -20\\ m\in \left[-20,\infty \right)\end{array}$

The solution of the inequality $\left|x\right|\le a$ is $x\ge -a$ and $x\le a$.

That implies the solution of the inequality $\left|x\right|\le a$ is the intersection of the solutions of the inequalities $x\ge -a$ and $x\le a$.

Find the intersection of the solutions of the inequalities $m+19\le 1$ and $m+19\ge -1$ to find the solution of the inequality $\left|m+19\right|\le 1$.

The intersection of the solutions of the inequalities $m+19\le 1$ and $m+19\ge -1$ is:

$m\in \left[-20,-18\right]$

Therefore, the solution of the inequality $\left|m+19\right|\le 1$ is $m\in \left[-20,-18\right]$.

Step 2. Draw the graph of the solution set which is m ∈ [ − 20 , − 18 ] .

The graph of the solution set which is $m\in \left[-20,-18\right]$  is: