Download Solving Nonlinear Inequalities

Section 2-7
Solving Nonlinear
Inequalities
Solution to Inequality
Equation
One solution
Inequality
Infinite
Solutions
2
Graphing Inequalities


0
[ , ] – number is included
( , ) – number is not included
( , ) – always used with

,

3
The solution set of an inequality is the set of all solutions.
Study the graph of the solution set of x2 + 3x  4  0.
[
-6 -5 - 4 -3 -2 -1
]
0
1
2
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
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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.
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Solve:
•
•
•
•
•
2
x
- 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.
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Solve:
2
x
• x = 11 or x = -3
- 8x – 33 > 0
Solutions: (-∞, -3) U (11, ∞)
-3
11
Test (-5)
Test (0)
Test (15)
(-5)2 - 8(-5) – 33 > 0
02 - 8(0) – 33 > 0
152 –8(15) – 33 > 0
25 + 40 – 33 > 0
-33 > 0
225 – 120 – 33 > 0
65 – 33 > 0
NO
105 – 33 > 0
32 > 0
72 > 0
YES
YES
7
Solve and graph :
x6
20
x2
x  6 2( x  2)

0
x2
x2
x  6  2x  4
0
x2
x6
2
x2
 x  10
0
x2
Critical Numbers : 2, 10
(2, 10]
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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,  )
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Homework
• WS 4-5
• Quiz next class
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