Download Solve Quadratic Inequalities

Math 35 (Spring ’09)
8.5 "Quadratic and Nonlinear Inequalities"
Objectives:
*
Solve quadratic and rational inequalities
*
Graph nonlinear inequalities in two variables
Preliminaries:
We have previously solve linear inequalities in one variable such as 2x + 3 > 8 and 6x 7 < 4x 9: To …nd their
solution sets, we used properties of inequalities to isolate the variable on one side of the inequality. In this section, we will
solve quadratic inequalities in one variable such as x2 + x 6 < 0 and x2 + 4x 5. We will use an interval testing method
on the number line to determine their solution sets.
Solve Quadratic Inequalities
Quadratic Inequalities:
A quadratic inequality can be written in one of the standard forms (where a; b; and c are real numbers and a 6= 0)
Solving Quadratic Inequalities:
Step 1 : Write the inequality in standard form and solve its related quadratic equation.
Step 2 : Locate the solutions (critical numbers) of the related quadratic equation on a number line.
Step 3 : Test each interval on the number line created in step 2 by choosing a test value from the interval.
The solution set includes the interval(s) whose test value makes the inequality true.
Step 4 : Determine whether the endpoints of the interval are included in the solution set.
Example 1: (Solving quadratic inequalities)
a)
x2 3x 4 > 0
b)
Page: 1
x2 + 2x
8<0
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
c)
x2
8x
8.5
15
d)
x2
9
Solve Rational Inequalities
9
Rational inequalities in one variable such as
< 8 and
x
method.
x2 + x
2
x
4
0 can be solved using the interval testing
Solving Rational Inequalities:
Step 1 : Write the inequality in standard form with a single quotient on the left side and 0 on the right side.
Step 2 : Set the numerator and denominator equal to zero and solve that equation.
Step 3 : Locate the solutions (critical numbers) found in step 2 on a number line.
Step 4 : Test each interval on the number line created in step 3 by choosing a test value from the interval.
The solution set includes the interval(s) whose test value makes the inequality true.
Step 5 : Determine whether the endpoints of the interval are included in the solution set.
Exclude any values that make the denominator 0.
Example 2: (Solving rational inequalities)
1
<3
a)
x
b)
Page: 2
x2
x + 2
2x
3
0
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
c)
x2 + x
6
x
4
0
8.5
d)
2
1
>
x + 1
x
Graph Nonlinear Inequalities in Two Variables
Example 3: (Graphing nonlinear inequalities)
a)
y > x2 3
y
-6
-4
b)
y
jx
y
6
6
4
4
2
2
-2
2
-2
4
6
-6
x
-4
-2
2
-2
-4
-4
-6
-6
Page: 3
2j
4
6
x
Notes by Bibiana Lopez