Sullivan Algebra and Trigonometry: Section 4.5 Solving Polynomial and Rational Inequalities

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Sullivan Algebra and Trigonometry Section
4.5Solving Polynomial and Rational Inequalities

• Objectives
• Solve Polynomial Inequalities
• Solve Rational Inequalities

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Steps for Solving Polynomial and Rational
Inequalities
1. Write the inequality so that the polynomial or
rational expression is on the left and zero is on
the right.
2. Determine when the expression on the left is
equal to zero or, with a rational expression, is
undefined.
3. Use the numbers found in step 2 to separate
the real number line into intervals.
4. Select a number in each interval and determine
if the function on the left is positive (gt 0) or
negative (lt 0). If the inequality isnt strict,
include the solutions to f(x) 0 in the
solution set.
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Example Solve the inequality x(x1) gt 20.
So, divide the number line at x 5 and at x
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Choose x 6, then 6( 6 1) gt 20 is TRUE.
So, this region is included in the solution set.
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Choose x 0, then 0(01) gt 20 is FALSE.
So, this region is not included in the solution
set.
Choose x 5, then 5(51) gt 20 is TRUE.
So, this region is included in the solution set.
Since this is a strict inequality, we do not
include the endpoints in the solution set. The
solution set is
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Example Solve the inequality
So, divide the number line at x 7 and at x
4. Note that 4 is NOT in the solution set.
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So, this region is included in the solution set.
Choose x 0, then 2(0) 3 gt 1
(0) 4
So, this region is not included in the solution
set.
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Choose x 5, then 2(5) 3 gt 1
(5) 4
So, this region is included in the solution set.
Since this is not a strict inequality but
involves an equal sign, we do include the
endpoint x 7 in the solution set, since the
left hand side is zero at x 7. The other
endpoint, x 4, cannot be included since it
makes the denominator equal to zero. The solution
set is