Question

Solve the inequality. Then graph its solution. $$|x+5| \leq 14$$

Short answer

The solution for the inequality \(|x+5| \leq 14\) is \(-19 \leq x \leq 9\). So, the graph is a solid line between -19 and 9 with both -19 and 9 being filled, indicating that those numbers are included in the solution.

Step-by-step solution

Set up the ‘positive’ equation

Start by keeping the expression inside the absolute bars positive. This involves keeping \(x+5\) as it is and removing the absolute value bars. The inequality \(x+5 \leq 14\) appears.

Solve the ‘positive’ inequality

This inequality can be solved simple algebra. Subtract 5 from each side of the inequality. After doing that, you will get \(x \leq 9\).

Set up the ‘negative’ equation

The second scenario involves taking the negation of the expression within the absolute value, which means flipping the sign of \(x+5\), resulting in \(-(x+5)\). As a result, the inequality becomes \(-x - 5 \leq 14\).

Solve the ‘negative’ inequality

To solve this inequality, do algebra. Add 5 to each side of the equation first, which results in \(-x \leq 19\). Realize that when you multiply or divide an inequality by a negative number, you have to flip the direction of the inequality sign. Multiply each side by -1, remembering to flip the inequality sign to get \(x \geq -19\).

Combine the results

The results from both the positive and negative scenarios will overlap. The solution will be all \(x\) values that make \(\(x \leq 9\) and \(x \geq -19\)\) true, meaning \(-19 \leq x \leq 9\). This implies any number between -19 and 9, including these two numbers, is a solution for the inequality.

Draw the graph

Draw a number line and mark both 9 and -19 on it. As both 9 and -19 are included solutions, the line drawn should be solid between them. The points -19 and 9 should be filled or closed circles.