1'8 Quadratic and Rational Inequalities MAC 1140 Mrs' Kessler

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1.8 Quadratic and Rational Inequalities
MAC 1140 Mrs. Kessler
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Definition of a Quadratic Inequality

• A quadratic inequality is any inequality that can
  be put in one of the forms
• ax2 bx c lt 0 ax2 bx c gt 0
• ax2 bx c lt 0 ax2 bx c gt 0
• where a, b, and c are real numbers and a 0.

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Procedure for Solving Quadratic Inequalities

• 1. Express the inequality in the standard form
• ax2 bx c lt 0 or ax2 bx c gt 0.
• 2. Solve the equation ax2 bx c 0. The real
  solutions are the boundary points.
• 3. Locate these boundary points on a number line,
  dividing the number line into test intervals.

x1
x2
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Procedure for Solving Quadratic Inequalities

• 4. Choose one representative number within each
  test interval.
• If substituting that value into the original
  inequality produces a true statement, then all
  real numbers in the test interval belong to the
  solution set.
• If substituting that value into the original
  inequality produces a false statement, then no
  real numbers in the test interval belong to the
  solution set.
• 5. Write the solution set the interval(s) that
  produced a true statement.

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Example 1
Solve and graph the solution set on a real number
line 2x2 3x
gt 2.
Step 1 Write the inequality in standard form. We
can write by subtracting 2 from both sides to get
zero on the right.
2x2 3x 2 gt 0
Step 2 Solve the related quadratic equation.
Replace the inequality sign with an equal
sign. 2x2 3x 2 0
(2x 1)(x 2) 0
x -1/2 or x 2
The boundary points are x 1/2 and 2.
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Example cont.
Solve and graph the solution set on a real number
line 2x2 3x gt 2.
Step 3 Locate the boundary points on a number
line. The number line with the boundary points
is shown as follows
The boundary points divide the number line into
three test intervals. Including the boundary
points (because of the given greater than or
equal to sign), the intervals are
(-ºº, -1/2, -1/2, 2, 2, ºº).
A
B
C
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Example 1 cont.
Solve and graph the solution set on a real number
line 2x2 3x gt 2.
C
A
B
Step 4 Take one representative number within each
test interval and substitute that number into the
original inequality. Test for positive, negative
(2x 1)(x 2) gt 0
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Example 1 cont.
Book way. Direct substitution
A
B
C
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Example cont.
2x2 3x gt 2.
Step 5 The solution set are the intervals that
produced a true statement. Our analysis shows
that the solution set is (-ºº, -1/2 U 2, ºº).
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Example 2 What about rational
inequalities?
1. Find the numbers that make the numerator and
denominator 0.
x -1 and x -3 These are the boundary points
2. Plot them on a number line divide the number
line into regions
-1
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Solve
Test a number in each region to see if it is
true or false.
What about the end points?
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Solve
B
(-3, 1)
-3
-1
Can we check this with a graphing
calculator? Graph y1 (x1)/(x3)
Where is the graphing under the x- axis?
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Look at the graph. Where is it lt 0?
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What if the problem were this?
B
-3
-1
(- 3, 1
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Example 3
Step 1 Express the inequality so that one side is
zero and the other side is a single quotient. We
subtract 2 from both sides to obtain zero on the
right. Add the fractions.
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Example 3 cont.
Step 2 Find boundary points by setting the
numerator and the denominator equal to zero.
x -7 or x -1
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Example 3 cont.
Step 3 Locate boundary points on a number line.
Test.
x -7 or x -1
(-?, -7 U (-1, ?)
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The Position Formula for a Free-Falling Object
Near Earths Surface

• An object that is falling or vertically projected
  into the air has its height in feet above the
  ground given by
• s -16 t 2 v0 t s0
• where s is the height in feet, v0 is the original
  velocity (initial velocity) of the object in feet
  per second, t is the time that the object is in
  motion in seconds, and s0 is the original height
  (initial height) of the object in feet.

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Example
An object is propelled straight up from ground
level with an initial velocity of 80 ft/sec. Its
height at time t is described by s -16 t 2
80 t where the height, s, is measured in feet
and the time, t, is measured in seconds. In which
time interval will the object be more than 64
feet above the ground?
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Example cont.
An object is propelled straight up from ground
level with an initial velocity of 80 fps. Its
height at time t is described by s -16 t 2 80
t where the height, s, is measured in feet and
the time, t, is measured in seconds. In which
time interval will the object be more than 64
feet above the ground?
-16 t 2 80 t gt 64
t 4
t 1
Since neither boundary point satisfy the
inequality, 1 and 4 are not part of the solution.
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Using help from Technology

• Where is the parabola
• above the blue line,
• y 64?

between 1 and 4 seconds
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Example cont. using test methods
With test intervals (-ºº, 1), (1, 4), and (4,
ºº), we could use 0, 2, and 5 as test points for
our analysis.
The object will be above 64 feet between 1 and 4
seconds.
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In class problems

• 1. x2 x - 6 gt 0

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In class problems

• 2. 6x2x gt 1

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In class problems
3.
Boundary points
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In class problems