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Chapter 9: Q. 384 (page 977)

In the following exercises, solve each inequality algebraically and write any solution in interval notation.
2x2+5x-12>0

Short Answer

Expert verified

The solution in interval notation is (-鈭�,-4)鈭�(1.5,鈭�).

Step by step solution

01

Step 1. Given information.

The given inequality is:

2x2+5x-12>0

02

Step 2. Determine the critical points.

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

2x2+5x-12=0

Split the middle term.

2x2+8x-3x-12=02x(x+4)-3(x+4)=0(x+4)(2x-3)=0x=-4,1.5

The two critical points -4and 1.5 divide the number line is three intervals (-鈭�,-4),(-4,1.5),(1.5,鈭�).

03

Step 3. Determine the intervals where the inequality is correct.

Test point for the interval (-鈭�,-4)is x=-5.

2(-5)2+5(-5)-12=13>0

Test point for the interval (-4,1.5)is x=0.

2(0)2+5(0)-12=-12<0

Test point for the interval localid="1645435817037" (1.5,鈭�)is x=2.

2(2)2+5(2)-12=6>0

The inequality 2x2+5x-12>0is true over the intervals (-鈭�,-4)and (1.5,鈭�).

04

Step 4. Conclusion.

The solution in interval notation is (-鈭�,-4)鈭�(1.5,鈭�).

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