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Chapter 2: Problem 16

Solve the inequality. Then graph the solution set. $$(x-3)^{2} \geq 1$$

Short Answer

Expert verified
The solutions to \((x-3)^2 \geq 1\) are \(x \geq 4\) and \(x \leq 2\). When graphed, there are two disjoint sections on the number line: one extending left from 2 (including 2), and one extending right from 4 (including 4).

Step by step solution

01

Simplify the inequality

Solving inequalities involving quadratics often requires breaking down the problem into two parts. In this case, \((x-3)^2 \geq 1\) becomes two inequalities: \(x - 3 \geq 1\) and \(x - 3 \leq -1\)
02

Solve the inequalities

For the first inequality, add 3 to both sides: \(x \geq 4\). For the second inequality, add 3 to both sides: \(x \leq 2\).
03

Plot the solution set

The solution to the inequality consists of two parts, \(x \geq 4\) and \(x \leq 2\), which refer to two disjoint sets of numbers. Therefore, when graphing these on a number line, use a closed dot to represent the inclusive inequality ('equal to' is included i.e., \(x\) can be 2 or 4), with a line extending right from 4 (as \(x\) is greater than or equal to 4) and a line extending left from 2 (as \(x\) is less than or equal to 2). The space between 2 and 4 is left empty as those numbers are not included in the solution set.

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Most popular questions from this chapter

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