Session 8
Agenda:
• Questions from 3.4-3.6?
• 4.1 – Solving Absolute Value Inequalities
• 4.2 – Solving Quadratic Inequalities
• 4.3 – Solving Rational Inequalities
• Things to do before our next meeting.
Questions?
4.1 – Solving Absolute Value Inequalities
•
Just as with solving absolute value equations, remember that
the absolute value of a number is its distance from the origin on
the number line. If we wanted to solve the inequality |x|≤ 4 we
want to find all values of x whose distance from the origin is
less than or equal to 4.
|
|
|
4
0
4
 4 x 4
•
In general, the solution to the inequality|x|≤ c, where c>0, is
–c≤x≤c which can be expressed in interval notation as [-c, c].
•
Similarly, the solution to the inequality|x|< c, where c>0, is
–c<x<c which can be expressed in interval notation as (-c, c).
•
If we wanted to solve the inequality |x|≥ 4, we want to find all
values of x whose distance from the origin is greater than or
equal to 4.
x  4
|
|
|
4
0
4
x4
•
In general, the solution to the inequality |x| ≥ c, where c>0, is
x≤-c or x≥c which can be expressed in interval notation as
(-∞, -c]  [c, ∞).
•
In general, the solution to the inequality |x| > c, where c>0, is
x<-c or x>c which can be expressed in interval notation as
(-∞, -c)  (c, ∞).
• Recall that absolute values are always non-negative.
Thus, an inequality of the form |x|>-5 is satisfied by all
values of x and the solution is (-∞, ∞). Conversely, an
inequality of the form |x|<-7 is not satisfied by any
values of x, and the inequality has no solution.
• When the expression inside the absolute value is more
complicated, use the general principles described in the
previous slides and solve the inequalities for x.
• Remember that when solving these inequalities, if you
multiply or divide both sides by a negative number, the
inequality sign flips.
Solve the following inequalities. Express your answers in
interval notation.
2x  5  7
3 10  7 x  5  4
3  2 x  11  7
6x  4  2  x
3  5x  2x  7
3x  9  3x  2
4.2 – Solving Quadratic Inequalities
•
To solve a quadratic inequality, the general strategy is to get all terms
to one side of the inequality, factor the quadratic expression, and use
a sign chart to determine on which intervals the inequality is satisfied.
•
The sign chart is divided into intervals based on where each factor is
equal to 0. Use a test point in each interval to determine the sign of
the quadratic on that interval.
•
For example, for the quadratic below, the sign chart is divided by -4
and 1. The test points x=-6, x=0, and x=2 were used to determine
the sign of the quadratic on that interval.
|
|
2
x  6
x  0
x  2
x  3x  4
(  )(  )  4
(  )(  )
1 (  )(  )
2
x  3x  4  0
( x  4)( x  1)  0
•
 

The solution to this inequality, then, is (  ,  4]  [1,  ) since these
are the intervals where the quadratic is greater than or equal to 0.
|
(  )(  )
4
|
(  )(  )
1 (  )(  )
 

• If the inequality was x 2  3 x  4  0, then the solution
would be (  ,  4)  (1,  ) .
• If the inequality was x 2  3 x  4  0 , then the solution
would be[  4, 1] .
2
• If the inequality was x  3 x  4  0, then the solution
would be (  4, 1) .
Solve the following inequalities. Express your answers in
interval notation.
x   14 x  45
2
3 x  11 x  4  0
2
2x  3x  1
2
x  81  18 x
2
9 x  12 x  4  0
2
4.3 – Solving Rational Inequalities
•
To solve a rational inequality, move ALL terms to one side of the
inequality (with 0 on the other side), simplify, and factor. Then,
use a sign chart as before to determine the intervals on which
the inequality is satisfied.
•
DO NOT multiply or divide both sides of the inequality by an
expression that involves x. Why? If x is positive the inequality
would remain the same, but if x is negative, the inequality
would flip.
|
|
•
Example:
x  4
x2
( x  5)
2
0
()
( )
x0
2
( )
( )
 
•
x6
5
( )
( )

The solution is [  2, 5)  (5,  ) . Note that x=5 is NOT included
in the solution since the expression is not defined there.
Solve the following inequalities. Express your answers in
interval notation.
x  x6
2
x6
0
x (4  x )
2
3 x  22 x  7
2
0
4x  3
x8
1
11 x  22
x 9
2
 2 x
4x
x4

x
2
x 1
Things to Do Before Next Meeting:
• Work on Sections 4.1-4.3 until you get all green
bars!
• Write down any questions you have.
• Continue working on mastering 3.4-3.6. After
you have all green bars on 3.1-3.6, retake the
Chapter 3 Test until you obtain at least 80%.
• Make sure you have taken the Chapter 5 Test
before our next meeting.