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Chapter 9: Q 9.131 (page 973)

Solve the following quadratic inequality algebraically and write the solution in interval notation.
x2+2x-8鈮�0

Short Answer

Expert verified

The solution, in interval notation, is(-鈭�,-4]鈭�[2,鈭�)

Step by step solution

01

Step 1. Given Information

The inequality in standard form isx2+2x-8鈮�0.

Change the inequality sign to an equal sign and then solve the equation.

x2+2x-8=0x2+4x-2x-8=0x+4x-2=0x=-4,x=2

02

Step 2. Test point

Use the critical points -4and 2 to divide the number line into intervals.

The above number line shows the sign of each quadratic expression using test points from each interval.

Take test points x=-5,x=0,x=5

03

Step 3. Check signs of inequality

Check the inequality sign for the above test points.

Forx=-5

x2+2x-8=(-5)2+2(-5)-8=25-10-8=7

Forx=0,

x2+2x-8=(0)2+2(0)-8=-8

For x=5,

x2+2x-8=(5)2+2(5)-8=25+10-8=27

04

Step 4. Solution of the inequality in interval notation

The inequality is positive in the first and last intervals.

The expression equals 0 at the points -4 and 2.

The solution, in interval notation, is(-鈭�,-4]鈭�[2,鈭�)

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