Transcript Solving Absolute Value Inequalities and Compound Inequalities

Solving Compound
Inequalities
Solving Absolute Value Inequalities
Example 1
x  3  6 or x  7  1
3 3
7 7
x3
x 8
3
5
8
This is a compound inequality. It
is already set up to start solving
the separate equations.
Since it has an “or” between the
two, just put both graphs on the
final graph and write your answer
in interval notation.
Solving Absolute Value Inequalities
Example 2
( 2) x
3 x  6 or  4 ( 2)
2
3
3
x  2
x 8
2 0
8
This is a compound inequality. It
is already set up to start solving
the separate equations.
Since it has an “or” between the
two, just put both graphs on the
final graph and write your answer
in interval notation.
Solving Absolute Value Inequalities
Example 3
3x  6  6 and 2x  8  0
8 8
6 6
3x  12
3 3
x4
4
2x  8
2
2
x  4
0
4
This is an and problem.
With and problems, we need to
find out where the two graphs
intersect. That will be the answer.
Draw both, and see where they
cross.
Solving Absolute Value Inequalities
Example 4
12  3x  3  9
3
3 3
 9  3x  12
3 3 3
3  x  4
3 0 4
[3,4)
This is another type of compound
inequality.
Whatever you do to get the x by
itself in the middle, you have to
do it to all “sides” of the
inequality.
Since it is written with two
inequalities in one sentence, it is
understood to have an “and”
between them. Therefore, solve,
and find the intersection.
Solving Absolute Value Inequalities
Practice.
Answers:
1)2 x  5  9 or 8  x  2
x
2)x  3  15 or  3  5
2
3)x  3  12 and x  4  1
4) 8  2 x  6  4
Solving Absolute Value Inequalities
Example 5
Write the inequality that fits the
given graph.
x  0 or x  4
0
2
4
Solving Absolute Value Inequalities
Example 6
Write the inequality that fits the
given graph.
7  x  5
7
0
5
Homework:
Page 317/ 9-23 odd,
27-31 odd, 38-41
Solving Absolute Value
Equations
Solving Absolute Value Equations
Example 1
x3  5
x  3  5 OR x  3  5
3 3
33
x  8
x2
This problem has absolute value
bars in it. Anytime you see
absolute value bars in an
equation, you need to split the
problem into two different
problems.
The first equation is the exact as
the original except just erase the
absolute value bars.
For the second equation, just
change the sign of the other side.
Solving Absolute Value Equations
Example 2
3x  6  15
3x  6  15
6 6
3x  9
x3
OR
3x  6  15
6
6
3x  21
x  7
Solving Absolute Value Equations
Example 3
2x 1  4  9
4 4
2 x  1  13
2x 1  13 OR 2x 1  13
1
1
1 1
2x  14
2 x  12
2
2
2
2
x7
x  6
Always make sure that the
absolute value bars are alone
first, so add the four to both sides
before you split it into two.
Solving Absolute Value Equations
Example 3.5
 3 x  2  9
3
3
x2 3
You cannot distribute numbers
into the absolute values. Since
the negative three is being
multiplied times the absolute
value bars, to get rid of them, we
x

2


3
x  2  3 OR
 2  2 need to divide both sides by the
2 2
negative three.
x5
x  1
Solving Absolute Value Equations
Example 4
x  4  3
Because the absolute values can
never equal a negative, there is
no work involved on this problem.
Homework:
Page 325/ 7-19 odd, 20
Bellwork
Practice.
Answers:
1)3 x  3  9 or 7  x  1
x
2)x  4  1 or  3  1
3
3)2 x  3  11 and x  4  1
4) 10  2 x  4  8
Solving Absolute Value
Inequalities and
Compound Inequalities
Solving Absolute Value Inequalities
Example 6
x2 5
x  2  5 or x  2  5
2 2
2
2
x  7
x3
With “or”, just put both
inequalities on the final
graph.
7
0
3
Because there is an absolute
value in the problem, that tells
me that I have to split the
problem into two pieces.
When you write it the second
time, not only do you change the
sign, but you also turn the
inequality around.
To decide if you use “and” or “or”,
remember GO to LA.
Greater than
Or
Less than
And
Solving Absolute Value Inequalities
Example 7
x2  6
x  2  6 and x  2  6
2 2
2
x  4
x 8
With “and”, find where
the two inequalities
intersect, and put that
on the final graph.
4
2
0
8
Because there is an absolute
value in the problem, that tells
me that I have to split the
problem into two pieces.
To decide if you use “and” or “or”,
remember GO to LA.
Greater than
Or
Less than
And
When you write it the second
time, not only do you change the
sign, but you also turn the
inequality around.
Solving Absolute Value Inequalities
Example 8
2x  4  8
2x  4  8 or 2x  4  8
2x  12
2x  4
x6
x  2
2
0
6
Solving Absolute Value Inequalities
Example 9
5x  4  8

Example 10
3x  7  2
When there is a negative on the
other side of an absolute value
inequality, the answer is either
“no solution” or “all real
numbers”.
Because the absolute value will
always be positive, if it is a
greater than, it will be “all real
numbers”.
If there is a less than sign with
the negative on the outside, the
answer is “no solution”.
Solving Absolute Value Inequalities
Example 11
6  3x  12
and
6  3x  12
 3x  6
x  2
2
0
6
6  3x  12
 3x  18
x6
Solving Inequalities
Example 12
2x  3  9
3 3
2x  6
2
2
x3
2
3
4
YOU DO NOT BREAK THIS
INTO TWO PROBLEMS
BECAUSE THERE ARE NO
ABSOLUTE VALUE BARS!!!
Homework:
Page 331/ 1-19 odd,
Page 119/ 1-19 odd