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Chapter 9: Q. 380 (page 977)

In the following exercises, solve each inequality algebraically and write any solution in interval notation.
x2+6x<-3

Short Answer

Expert verified

The solution in interval notation is (-3-6,-3+6).

Step by step solution

01

Step 1. Given information.

The given inequality is:

x2+6x<-3

02

Step 2. Determine the critical points.

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

x2+6x=-3x2+6x+3=0

Use the quadratic formula to solve the equation.

x=-(6)卤(6)2-4(1)(3)2(1)鈭�x=-b卤b2-4ac2ax=-6卤242x=-62卤262x=-3卤6

The two critical points -3-6and -3+6divide the number line is three intervals (-鈭�,-3-6),(-3-6,-3+6),(-3+6,鈭�).

03

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

Test point for the interval (-鈭�,-3-6)is x=-6.

(-6)2+6(-6)=0>-3

Test point for the interval (-3-6,-3+6)is x=-1.

(-1)2+6(-1)=-5<-3

Test point for the interval (-3+6,鈭�)is x=0.

(0)2+6(0)=0>-3

The inequality x2+6x<-3is true over the interval (-3-6,-3+6).

04

Step 4. Conclusion.

The solution in interval notation is (-3-6,-3+6).

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