Question

Solve the inequality. Then graph the solution. $$|x-3| \leq 17$$

Short answer

The solution to the absolute value inequality \(|x - 3| ≤ 17\) is \(-14 ≤ x ≤ 20\). All numbers within and including -14 and 20 on a number line represent this solution.

Step-by-step solution

Understand the inequality

The inequality \(|x - 3| ≤ 17\) states that the absolute difference between x and 3 should be less than or equal to 17. This means that x could be at most 17 units away from 3 in either direction. Thus, \(x - 3\) could be anything between -17 and 17.

Solve the inequality

To turn the absolute value inequality into a normal inequality, you can understand it as two separate inequalities: \(x - 3 ≤ 17\) and \(x - 3 ≥ -17\). Solving these two inequalities separately, you get the solutions \(x ≤ 20\) and \(x ≥ -14\).

Find the intersection

The solution to the original inequality must satisfy both of the inequalities from step 2 at the same time. The range of x-values that satisfy both is \(-14 ≤ x ≤ 20\). This can be found by identifying the intersection of the solutions to both inequalities.

Graph the solution

On a number line, you'd mark the numbers -14 and 20. Because the inequality is less than or equal to -14 and 20, you'll have closed circles on these points. Draw a line between them to denote every number in between these two is included in the solution. The solution on the number line is every value between -14 and 20 inclusive.