Question

Solve the inequality. Then graph the solution. $$|x+3| \geq 8$$

Short answer

The solutions to the inequality are \(x \leq -11\) and \(x \geq 5\). In interval notation, the solution set is \((-\infty, -11] \cup [5, \infty )\).

Step-by-step solution

Solve for x when the quantity inside the absolute value is positive

Here, we solve the inequality assuming that the quantity inside the absolute value brackets is positive. This yields the inequality \(x+3 \geq 8\). Solving for x, we subtract 3 from both sides, leaving us with the inequality \(x \geq 5\).

Solve for x when the quantity inside the absolute value is negative

Here, we solve the inequality assuming that the quantity inside the absolute value brackets is negative. This gives us the inequality \(-(x+3) \geq 8\). We distribute the negative sign and swap the sides to keep the inequality sign correct, yielding \(8 \geq x+3\). Solving for x, we subtract 3 from both sides, leading to the inequality \(x \leq -11\).

Graph the solution

On the number line, we include all numbers larger than or equal to 5. This means shading the section of the line to the right of 5. Also, we include all numbers less than or equal to -11. This means shading the section of the line to the left of -11. The interval notation for the solution set is \((-\infty, -11] \cup [5, \infty )\).