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1.8. OTHER TYPES OF INEQUALITIES
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• Solve polynomial inequalities.
• Solve rational inequalities.
• Use inequalities to model and solve real-life
problems.
What You Should Learn
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Polynomial Inequalities
To solve a polynomial inequality such as x2 – 2x – 3 0,
you can use the fact that a polynomial can change signs
only at its zeros (the x-values that make the polynomial
equal to zero).
Between two consecutive zeros, a polynomial must be
entirely positive or entirely negative. This means that when
the real zeros of a polynomial are put in order, they divide
the real number line into intervals in which the polynomial
has no sign changes.
These zeros are the key numbers of the inequality, and
the resulting intervals are the test intervals for the
inequality.
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Example
The quadratic polynomial factors as
x2 – 2x – 3 = (x + 1)(x – 3)
and has two zeros, x = –1 and x = 3.
These zeros divide the real number line into three test
intervals:
( , –1), (–1, 3), and (3, ).
Three test intervals for x2 – 2x – 3
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Polynomial Inequalities
So, to solve the inequality x2 – 2x – 3 0, you need only
test one value from each of these test intervals to
determine whether the value satisfies the original
inequality. If so, you can conclude that the interval is a
solution of the inequality.
You can use the same basic approach to determine the
test intervals for any polynomial.
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Example
Solve x2 – x – 6 0.
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Solution
By factoring the polynomial as
x2 – x – 6 = (x + 2)(x – 3 )
you can see that the key numbers are x = –2 and x = 3.
So, the polynomial’s test intervals are
( , –2), (–2, 3), and (3, ). Test intervals
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Solution In each test interval, choose a representative x-value and
evaluate the polynomial. You may choose any number you
want from each test interval.
Test Interval x-Value Polynomial Value Conclusion
( , –2) x = –3 (–3)2 – (–3) – 6 = 6 Positive
(–2, 3) x = 0 (0)2 – (0) – 6 = – 6 Negative
(3, ) x = 4 (4)2 – (4) – 6 = 6 Positive
From this you can conclude that the inequality is satisfied
for all x-values in (–2, 3).
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Solution
This implies that the solution of the inequality x2 – x – 6 0
is the interval (–2, 3).
Note that the original inequality contains a “less than”
symbol. This means that the solution set does not contain
the endpoints of the test interval (–2, 3).
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Remark
As with linear inequalities, you can check the
reasonableness of a solution by substituting x-values into
the original inequality.
For instance, to check the solution found in the previous
Example, try substituting several x-values from the interval
(–2, 3) into the inequality
x2 – x – 6 0.
Regardless of which x-values you choose, the inequality
should be satisfied.
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Rational Inequalities
The concepts of key numbers and test intervals can be
extended to rational inequalities.
To do this, use the fact that the value of a rational
expression can change sign only at its zeros (the x-values
for which its numerator is zero) and its undefined values
(the x-values for which its denominator is zero).
These two types of numbers make up the key numbers of a
rational inequality. When solving a rational inequality, begin
by writing the inequality in general form with the rational
expression on the left and zero on the right.
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Example
Solve
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Solution
Write original inequality.
Write in general form.
Find the LCD and subtract
fractions.
Simplify.
Key Numbers: x = 5, x = 8 Zeros and undefined values
of rational expression
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Solution
Key Numbers: x = 5, x = 8
Test Intervals: ( , 5), (5, 8), (8, ).
Test: Is
After testing these intervals, you can see that the inequality
is satisfied on the open intervals ( , 5), and (8, ).
Zeros and undefined values
of rational expression
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Solution
Moreover, because when x = 8, you can
conclude that the solution set consists of all real numbers
in the intervals ( , 5) [8, ).(Be sure to use a closed
interval to indicate that x can equal 8.)