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Solving Nonlinear Inequalities
Section 2-7
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2
Solution to Inequality
Equation
One solution
Inequality
Infinite Solutions
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3
Graphing Inequalities
0
[ , ] – number is included
( , ) – number is not included
( , ) – always used with ,
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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][
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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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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.
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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, ∞)
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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(
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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, )
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10
Homework• WS 4-5
• Quiz next class