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Chapter 2: Problem 4

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$(x+1)(x-7) \leq 0$$

Short Answer

Expert verified
The solution to the inequality \((x+1)(x-7) \leq 0\) is \([-1,7]\) in interval notation.

Step by step solution

01

Find the Roots

The roots of the inequality \((x+1)(x-7) \leq 0\) are found by setting each factor equal to zero. Those factors are \(x+1\) and \(x-7\). Solving \(x+1=0\) gives \(x=-1\) and solving \(x-7=0\) gives \(x=7\).
02

Determine the Test Intervals

With these roots, the number line can be divided into three intervals. These intervals are \((-鈭�,-1)\), \((-1,7)\), and \((7,鈭�)\).
03

Test the Intervals

Choose a test value from each interval and substitute it into the inequality to see if it makes the inequality true. For \((-鈭�,-1)\), let's choose \(x=-2\). Then \((-2+1)(-2-7) = 9\) which is more than 0, so this interval does not satisfy the inequality. For \((-1,7)\), let's choose \(x=0\). Then \((0+1)(0-7) = -7\) which is less than or equal to zero, so this interval satisfies the inequality. For \((7,鈭�)\), let鈥檚 choose \(x=8\). Then \((8+1)(8-7) = 8\), which is more than 0 so this interval also does not satisfy the inequality.
04

Write the Solution in Interval Notation

In interval notation, the solution set of the inequality \((x+1)(x-7) \leq 0\) is \([-1,7]\), including the endpoints -1 and 7 since the original inequality allows for equality (鈮� rather than <).

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