Q33. - Algebra 1

Question

Solve each compound inequality. Then graph the solution set.

$3x+2\le 11$ or $5x-8>22$

Step-by-Step Solution

Answer

The solution of the given compound inequality $3x+2\le 11$ or $5x-8>22$ is $x\in \left(-\infty ,3\right]\cup \left(6,\infty \right)$.

The graph of the solution set which is $x\in \left(-\infty ,3\right]\cup \left(6,\infty \right)$ is:

Step 1. Solve the given compound inequality 3 x + 2 ≤ 11 or 5 x − 8 > 22 .

The solution of the given compound inequality $3x+2\le 11$ or $5x-8>22$ is:

Solve the inequality $3x+2\le 11$.

$$\begin{gathered} 3x+2\le 11 \\ 3x+2-2\le 11-2 \\ 3x\le 9 \\ \frac{3x}{3}\le \frac{9}{3} \\ x\le 3 \\ x\in \left(-\infty ,3\right] \end{gathered}$$

Solve the inequality $5x-8>22$.

$$\begin{gathered} 5x-8>22 \\ 5x-8+8>22+8 \\ 5x>30 \\ \frac{5x}{5}>\frac{30}{5} \\ x>6 \\ x\in \left(6,\infty \right) \end{gathered}$$

A compound inequality containing ‘or’ is true if one or both inequalities are true.

That implies the solution of the compound inequality containing ‘or’ is the union of the solutions of the two simple statements.

Find the union of the solutions of the inequalities $3x+2\le 11$ and $5x-8>22$ to find the solution of the compound inequality $3x+2\le 11$ or $5x-8>22$.

The union of the solutions of the inequalities $3x+2\le 11$ and $5x-8>22$ is:

$x\in \left(-\infty ,3\right]\cup \left(6,\infty \right)$

Therefore, the solution of the compound inequality $3x+2\le 11$ or $5x-8>22$ is $x\in \left(-\infty ,3\right]\cup \left(6,\infty \right)$.

Step 2. Draw the graph of the solution set which is x ∈ − ∞ , 3 ∪ 6 , ∞ .

The graph of the solution set which is $x\in \left(-\infty ,3\right]\cup \left(6,\infty \right)$ is: