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Chapter 9: Q. 379 (page 977)

In the following exercises, solve each inequality algebraically and write any solution in interval notation.
x2-10x>-19

Short Answer

Expert verified

The solution in interval notation is (-鈭�,5-6)鈭�(5+6,鈭�).

Step by step solution

01

Step 1. Given information.

The given inequality is:

x2-10x>-19

02

Step 2. Determine the critical points.

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

x2-10x=-19x2-10x+19=0

Use the quadratic formula to solve the equation.

x=-(-10)卤(-10)2-4(1)(19)2(1)鈭�x=-b卤b2-4ac2ax=10卤242x=102卤262x=5卤6

The two critical points 5-6and 5+6divide the number line is three intervals (-鈭�,5-6),(5-6,5+6),(5+6,鈭�).

03

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

Test point for the interval (-鈭�,5-6)is x=0.

(0)2-10(0)=0>-19

Test point for the interval (5-6,5+6)is x=4.

(4)2-10(4)=-24<-19

Test point for the interval (5+6,鈭�)is x=8.

(8)2-10(8)=-16>-19

The inequality x2-10x>-19is true over the intervals (-鈭�,5-6)and (5+6,鈭�).

04

Step 4. Conclusion.

The solution in interval notation is (-鈭�,5-6)鈭�(5+6,鈭�).

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