Solving Nonlinear Inequalities

Section 2-7

2

Solution to Inequality

Equation

One solution

Inequality

Infinite Solutions

3

Graphing Inequalities

0

[ , ] – number is included

( , ) – number is not included

( , ) – always used with ,

4

The solution set of an inequality is the set of all solutions.Study the graph of the solution set of x2 + 3x 4 0.

The solution set is {x | 4 x 1}.

The values of x for which equality holds are part of the solution set.

These values can be found by solving the quadratic equation associated with the inequality.

x2 + 3x 4 = 0 Solve the associated equation.

(x + 4)(x 1) = 0 Factor the trinomial.

x = 4 or x = 1 Solutions of the equation

-2 -1 0 1 2-6 -5 - 4 -3][

5

Vocabulary

• The Critical Numbers of any rational expression inequality are the zeros and undefined numbers.

• Critical Numbers for a quadratic inequality are the roots.

• These numbers will be used to establish the test intervals over which we will solve the inequalities.

• Zeros are where the numerator will be zero• Undefined Numbers are where the denominator

will be zero.

6

Solve: x2 - 8x – 33 > 0

• x2 - 8x – 33 > 0

• (x - 11) (x + 3) > 0

• x = 11 or x = -3 (These are critical numbers)

• Test 3 areas x < -3, -3 < x < 11, x > 11

• See next slide.

7

Solve: x2 - 8x – 33 > 0• x = 11 or x = -3

-3 11

Test (-5)

(-5)2 - 8(-5) – 33 > 0

25 + 40 – 33 > 0

65 – 33 > 0

32 > 0

YES

Test (0)

02 - 8(0) – 33 > 0

-33 > 0

NO

Test (15)

152 –8(15) – 33 > 0

225 – 120 – 33 > 0

105 – 33 > 0

72 > 0

YES

Solutions: (-∞, -3) U (11, ∞)

8

Solve and graph : 22

6

x

x

022

6

x

x

02

)2(2

2

6

x

x

x

x

02

426

x

xx

02

10

x

x

10,2:NumbersCritical

]10,2(

9

Unusual Solution Sets

• There are 4 cases of possible unusual solutions:

• 1. The solution can be all real numbers (- , )

• 2. The solution may be a single number {3}

• 3. The solution may be the Null Set, (No solution)

• 4. The solution may be all numbers except one, like (-, 3) U (3, )

10

Homework• WS 4-5

• Quiz next class