Question

Solve each inequality, graph the solution, and write the solution in interval notation.

$\frac{3}{4}x+2\le -1or\frac{1}{2}(x+8)\ge -3$

Short answer

Solution of inequalities

$x\le -4orx\ge -14$

Graph of solution of inequalities

The solution in interval notation

$(-\infty ,\infty )$

Step-by-step solution

Step 1. Given information 

Consider each inequality

$\frac{3}{4}x+2\le -1or\frac{1}{2}(x+8)\ge -3$

Step 2. Solve each inequality 

$$\begin{gathered} \frac{3}{4}x+2\le -1 \\ \frac{3}{4}x\le -1-2 \\ \frac{3}{4}x\le -3 \\ x\le -3\times \frac{4}{3} \\ x\le -4 \end{gathered}$$ or $$\begin{gathered} \frac{1}{2}(x+8)\ge -3 \\ x+8\ge -3\times 2 \\ x+8\ge -6 \\ x\ge -6-8 \\ x\ge -14 \end{gathered}$$

Step 3. Graph each inequality  

For$x\le -4$

For$x\ge -14$

Step 4. Graph number that make either one inequality true x≤-4 or x≥-14

The solution in interval notation

$(-\infty ,-4]\cup [-14,\infty )=(-\infty ,\infty )$