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Chapter 2: Problem 30

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$(x+1)(x+2)(x+3) \geq 0$$

Short Answer

Expert verified
The solution to the polynomial inequality \((x+1)(x+2)(x+3) \geq 0\) is \([-3, -2] \cup [-1, \infty)\).

Step by step solution

01

Identify the Critical Numbers

Set each term of the inequality \((x+1)(x+2)(x+3) \geq 0\) equals to zero and solve for \(x\). This gives us \(x = -1, -2, -3\). These are the critical numbers.
02

Set Up the Intervals

The critical numbers divide the real number line into four intervals: \(-\infty, -3\), \(-3, -2\), \(-2, -1\), and \(-1, \infty\).
03

Test Each Interval

Choose a test point from each interval (say -4, -2.5, -1.5, and 0) and see if that value will satisfy the inequality. Remember to respect the order of the factors when you evaluate the test points. If the test point satisfies the inequality, then all the numbers in that interval are solutions to the inequality.
04

Graph the Solution and Write It in Interval Notation

On the number line, show each interval where the inequality is satisfied. Since the inequality is '\(\geq\)', the critical numbers are included in the solution set. In interval notation, this solution set is written as \([-3, -2] \cup [-1, \infty)\).

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