Transcript 2.4 Solve Polynomial Inequalities

2.4
Solve Polynomial Inequalities
Inequalities Test: 2/26/10
Vocabulary
A polynomial inequality in one variable can
be written as one of the following:
1. anxn, an-1xn-1 +…+a1x + a0 < 0
2. anxn, an-1xn-1 +…+a1x + a0 > 0
3. anxn, an-1xn-1 +…+a1x + a0 < 0
4. anxn, an-1xn-1 +…+a1x + a0 > 0
where an = 0
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Vocabulary
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Inequalities can be used to describe subsets
of real numbers called intervals.
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In bounded intervals below, the real numbers a
and b are endpoints of each interval.
Inequality a<x<b
Notation
[a,b]
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a<x<b
(a,b)
a<x<b a<x<b
[a,b) (a,b]
Unbounded intervals are also written in this
notation
Inequality x > a
x>a
Notation [a, +∞) (a, +∞)
x<b
(-∞, b]
x<b
(-∞, b)
Example 1:
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Solve x3 – 3x2 > 10x algebraically
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Step 1: Rewrite as an equation, and put in
standard form.
Step 2: Solve for x (we can factor)
Example 1 Cont’d:
Plotting our found x-values will help us find
our intervals. This breaks our number line
into 4 intervals. Test an x-value in each
interval to see if it satisfies the inequality
Example 1 Cont’d:
(-2, 0)
(5, ∞)
WORKS
Does NOT work
WORKS
Does NOT work
You Try:
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1.
Solve the inequality algebraically
-3x3 + 10x2 < - 8x
Example 2:
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Solve 2x3 + x2 – 6x < 0 by graphing.
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Graph and find your zeros
x = -2, 0, 3/2
(-∞, -2] and [0, 3/2]
You Try:
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1.
Solve the inequality using a graph
2x3 + 8x2 < - 6x