2'6 Polynomial and Rational Inequalities

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2.6 Polynomial and Rational Inequalities

• Solve polynomial and rational inequalities.

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To solve a polynomial inequality

• (1) compare to zero
• (2) factor the polynomial and find the zeros.
  These are boundary points.
• (3) determine intervals where each factor is
  /-(use a line and test values)
• (4)summarize the results
• (5) use the results to solve the inequality

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Solve
Compare to 0
Factor
Determine the boundary points by setting each
factor 0, then solve.
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Example continued
Use these values to form intervals on a number
line.
3
-1
5
Pick a test value from each interval to determine
the sign of each factor in that interval
-1
3
4
-2
0
-
-

-


-
product


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Finally, since the product is positive in these
intervals our solutions are
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The graph can also be used to determine the
intervals that are solutions.
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Example

• Solve 4x3 ? 7x2 ? 15x.
• We need to find all the zeros of the function so
  we solve the related equation.
• The zeros are 0, 3 and ?5/4. Thus the
  x-intercepts of the graph are (0, 0), (3, 0) and
  (?5/4, 0).

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Example continued

• The zeros divide the x-axis into four intervals.
  For all x-values within a given interval, the
  sign of 4x3 ? 7x2 ? 15x ? 0 must be either
  positive or negative. To determine which, we
  choose a test value for x from each interval and
  find f(x).

0
3
-2
1
-1
4
result
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Since we are solving 4x3 ? 7x2 ? 15x ? 0, the
solution set consists of only two of the four
intervals, those in which the sign of f(x) is
negative. x ?? lt x lt ?5/4 or 0 lt x lt 3.
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Solve each inequality
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Rational inequalities are solved using a method
similar to quadratic inequalities. Recall that if
a product or quotient has an even number of
negative factors, then its value is positive and
if it has an odd number of negative factors then
its value is negative.
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Rational functions work using a very similar
approach.

• (1) Be sure to write the quotient as a single
  fraction compared to zero.
• (2) Factor the numerator and denominator.
• (3) Determine intervals where each factor is /-.
  (Make a table of values using the intervals found
  by setting each factor 0).

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4. Solve the inequality being careful to include
or exclude the endpoints. Remember that a
fraction equals zero when the numerator is zero
and that the denominator cannot equal zero
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Solve

Compare to 0
or
Write as a single fraction
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Find zeros
Test values
3
0
2
4





Result
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This gives solutions to
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Solve each inequality