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Chapter 9: Q. 383 (page 977)

In the following exercises, solve each inequality algebraically and write any solution in interval notation.
-2x2+7x+4鈮�0

Short Answer

Expert verified

The solution in interval notation is [-0.5,4].

Step by step solution

01

Step 1. Given information.

The given inequality is:

-2x2+7x+4鈮�0

02

Step 2. Determine the critical points.

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

-2x2+7x+4=0

Split the middle term.

-2x2+8x-x+4=0-2x(x-4)-1(x-4)=0-(2x+1)(x-4)=0x=-0.5,4

The two critical points -0.5and 4 divide the number line is three intervals (-鈭�,-0.5),(-0.5,4),(4,鈭�).

03

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

Test point for the interval (-鈭�,-0.5)is x=-1.

-2(-1)2+7(-1)+4=-5<0

Test point for the interval (-0.5,4)is x=0.

-2(0)2+7(0)+4=4鈮�0

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

-2(5)2+7(5)+4=-11<0

The inequality -2x2+7x+4鈮�0is true over the interval [-0.5,4].

04

Step 4. Conclusion.

The solution in interval notation is [-0.5,4].

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