Transcript Chapter 3 Section 7 - Canton Local Schools

Chapter 3
Polynomial and
Rational
Functions
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All rights reserved
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SECTION 3.7
Polynomial and Rational Inequalities
OBJECTIVES
1
Solve quadratic inequalities.
2
Solve polynomial inequalities.
3
Solve rational inequalities.
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USING TEST POINTS TO SOLVE
INEQUALITIES
In this case of quadratic polynomials, we find
the two roots (using the quadratic formula if
necessary) and then make sign graphs of the
quadratic polynomial.
The two roots divide the number line into
three intervals, and the sign of the quadratic
polynomial is determined in each interval by
testing any number in the interval.
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EXAMPLE 1
Using the Test-Point Method to Solve a
Quadratic Inequality
Solve x2 > x + 6. Write the solution in interval
notation and graph the solution set.
Solution
Rearrange the inequality so that 0 is on the
right side.
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EXAMPLE 1
Using the Test-Point Method to Solve a
Quadratic Inequality
Solution continued
Solve the associated equation.
The two roots are –2 and 3.These roots
divide the number line into three intervals
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EXAMPLE 1
Using the Test-Point Method to Solve a
Quadratic Inequality
Solution continued
Select a “test point” in each of the intervals.
Then evaluate x2 – x – 6 at each test point.
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EXAMPLE 1
Using the Test-Point Method to Solve a
Quadratic Inequality
Solution continued
The sign at each test point gives the sign for
every point in the interval containing that
test point.
The solution set is (–∞, –2)  (3, ∞).
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EXAMPLE 2
Calculating Speeds from Telltale Skid
Marks
A car involved in an accident left skid marks
over 75 feet long. Under the road conditions at
the accident, the distance (in feet) it takes a car
traveling miles per hour to stop is given by the
equation
d = 0.05v2 + v
The accident occurred in a 25-mile-per-hour
speed zone. Was the driver going over the speed
limit?
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EXAMPLE 2
Calculating Speeds from Telltale Skid
Marks
Solution
Solve the inequality
(stopping distance) > 75 ft
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EXAMPLE 2
Calculating Speeds from Telltale Skid
Marks
Solution continued
The two roots divide the number line into three
test intervals.
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EXAMPLE 2
Calculating Speeds from Telltale Skid
Marks
Solution continued
The results are shown in the sign graph.
Look at only the positive values of v. Note
that the numbers corresponding to speeds
between 0 and 30 mph are not solutions.
Thus, the car was traveling more than
30 mph. The driver was going over the
speed limit.
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ONE-SIGN THEOREM
If a polynomial equation has no real solution,
then the polynomial is always positive or
always negative.
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EXAMPLE 3
Using the One-Sign Theorem to Solve a
Quadratic Inequality
Solve x2 – 2x + 2 > 0.
Solution
Set the left side equal to 0 to obtain the
associated equation. Evaluate the
discriminant to see if there are any real roots.
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EXAMPLE 3
Using the One-Sign Theorem to Solve a
Quadratic Inequality
Solution continued
The discriminant is negative so there are no
real roots and x2 – 2x + 2 is always positive
or always negative.
Pick 0 as a test point.
is positive; so
x2 – 2x + 2 is always positive.
The solution set is (–∞, ∞).
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A PROCEDURE FOR SOLVING POLYNOMIAL
INEQUALITIES
P(x) > 0, P(x) ≥ 0, P(x) < 0 OR P(x) ≤ 0
Step 1 Arrange the inequality so that 0 is on
the right-hand side.
Step 2 Solve the associated equation P(x) = 0.
The real solutions of this equation
form the boundary points.
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A PROCEDURE FOR SOLVING POLYNOMIAL
INEQUALITIES
P(x) > 0, P(x) ≥ 0, P(x) < 0 OR P(x) ≤ 0
Step 3 Locate the boundary points on the
number line. These boundary points
divide the number line into intervals.
Write these intervals.
Step 4 Select a test point in each interval
from Step 3 and evaluate P(x) at each
test point to determine the sign of P(x)
on that interval.
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A PROCEDURE FOR SOLVING POLYNOMIAL
INEQUALITIES
P(x) > 0, P(x) ≥ 0, P(x) < 0 OR P(x) ≤ 0
Step 5 Write the solution set by selecting the
intervals in Step 4 that satisfy the
given inequality. If the inequality is
P(x) ≤ 0 (or ≥ 0), include the
corresponding boundary points.
Graph the solution set.
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EXAMPLE 4
Solving a Polynomial Inequality
Solve x3 + 2x + 7 ≤ 3x2 + 6x – 5.
Solution
1.
2.
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EXAMPLE 4
Solving a Polynomial Inequality
Solution continued
2. continued
The boundary points are –2, 2, and 3.
3.
The boundary points divide the number line
into the intervals: (–∞, –2), (–2, 2), (2, 3)
and (3, ∞).
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EXAMPLE 4
Solving a Polynomial Inequality
Solution continued
4.
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EXAMPLE 4
Solving a Polynomial Inequality
Solution continued
5. The solution set is (–∞, –2]  [2, 3].
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EXAMPLE 5
Solving a Polynomial Inequality
Solve (x – 1)(x + 1)(x2 + 1) ≤ 0.
Solution
2. Find the real solutions of the associated
equation.
The real solutions are 1 and –1. (x2 + 1 has
no real solution.
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EXAMPLE 5
Solving a Polynomial Inequality
Solution continued
3 and 4.
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EXAMPLE 5
Solving a Polynomial Inequality
Solution continued
5.
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EXAMPLE 6
Solve
Solving a Rational Inequality
Write the solution in
interval notation and graph the solution set.
Solution
1.
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EXAMPLE 6
Solving a Rational Inequality
Solution continued
2. Set the numerator and the denominator
of the left side equal to 0.
3. The numbers 1 and 4 divide the number
line into three intervals: (–∞, 1), (1, 4) and
(4, ∞).
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EXAMPLE 6
Solving a Rational Inequality
Solution continued
4.
5.
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EXAMPLE 7
Solving a Rational Inequality
Solve:
Solution
2.
x = –4, –2, 1, or 3
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EXAMPLE 7
Solving a Rational Inequality
Solution continued
3 and 4.
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EXAMPLE 7
Solving a Rational Inequality
Solution continued
3 and 4. continued
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EXAMPLE 7
Solving a Rational Inequality
Solution continued
5. The solution set is (–4, –2]  [1, 3).
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