Question

Solve the inequality. $$|x+3| < 8$$

Short answer

The solution to the inequality \(|x+3| < 8\) is \(-11 < x < 5\).

Step-by-step solution

Set Up Two Inequalities

To solve an inequality involving absolute value, it is necessary to set up two separate inequalities. The statement \(|x+3| < 8\) actually conveys two bits of information. They are: \(x+3 < 8\) and \(-(x+3) < 8\).

Simplify the Inequalities

Next, simplify the inequalities. This involves solving as one would any other algebraic equation. For the inequality \(x + 3 < 8\), subtract 3 from both sides of the inequality to isolate x, producing \(x < 5\). For the inequality \(-x - 3 < 8\), subtract 3 from both sides to get \(-x < 11\), then multiply each side by -1 to get \(x > -11\). Remember that the direction of the inequality sign changes when multiply or divided by a negative.

The Final Answer

To express the solution to the original inequality, we need to combine the solutions of the two inequalities, taking into the account that the x in the original inequality is the intersection of x in both our inequalities. So the final answer is \(-11 < x < 5\).