Download Solving Absolute Value Equations and Inequalities

Solving Absolute Value
Equations and Inequalities
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both
in the U.S. and worldwide. Using an open-content, web-based
collaborative model termed the FlexBook®, CK-12 intends to
pioneer the generation and distribution of high-quality educational
content that will serve both as core text as well as provide an
adaptive environment for learning, powered through the FlexBook
Platform®.
Copyright © 2014 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/terms.
Printed: September 18, 2014
www.ck12.org
Chapter 1. Solving Absolute Value Equations and Inequalities
C HAPTER
1
Solving Absolute Value
Equations and Inequalities
Objective
To solve equations and inequalities that involved absolute value.
Review Queue
Solve the following equations and inequalities
1. 5x − 14 = 21
2. 1 < 2x − 5 < 9
3. − 43 x + 7 ≥ 19
4. 4(2x + 7) − 15 = 53
Solving Absolute Value Equations
Objective
To solve absolute value equations.
Watch This
Watch the first part of this video.
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/94
Khan Academy: Absolute Value Equations
Guidance
Absolute value is the distance a number is from zero. Because distance is always positive, the absolute value will
always be positive. Absolute value is denoted with two vertical lines around a number, |x|.
|5|= 5
|−9|= 9
|0|= 0
|−1|= 1
1
www.ck12.org
When solving an absolute value equation, the value of x could be two different possibilities; whatever makes the
absolute value positive OR whatever makes it negative. Therefore, there will always be TWO answers for an absolute
value equation.
If |x|= 1, then x can be 1 or -1 because |1|= 1 and |−1|= 1.
If |x|= 15, then x can be 15 or -15 because |15|= 15 and |−15|= 15.
From these statements we can conclude:
Example A
Determine if x = −12 is a solution to |2x − 5|= 29.
Solution: Plug in -12 for x to see if it works.
|2(−12) − 5| = 29
|−24 − 5| = 29 X
|−29| = 29
-12 is a solution to this absolute value equation.
Example B
Solve |x + 4|= 11.
Solution: There are going to be two answers for this equation. x + 4 can equal 11 or -11.
|x + 4|= 11
.&
x + 4 = 11 x + 4 = −11
or
x=7
x = −15
Test the solutions:
|7 + 4|= 11
|11|= 11 X
|−15 + 4|= 11
|−11|= 11 X
Example C
2
Solve 3 x − 5 = 17.
Solution: Here, what is inside the absolute value can be equal to 17 or -17.
2
www.ck12.org
Chapter 1. Solving Absolute Value Equations and Inequalities
2
x − 5 = 17
3
.&
2
2
x − 5 = 17
x − 5 = −17
3
3
2
2
x = 22
or
x = −12
3
3
3
3
x = −12 ·
x = 22 ·
2
2
x = 33
x = −18
Test the solutions:
2
(33) − 5 = 17
3
|22 − 5|= 17 X
2
(−18) − 5 = 17
3
|−12 − 5|= 17 X
|17|= 17
|−17|= 17
Guided Practice
1. Is x = −5 a solution to |3x + 22|= 6?
Solve the following absolute value equations.
2. |6x − 11|+2 = 41
1
3. 2 x + 3 = 9
Answers
1. Plug in -5 for x to see if it works.
|3(−5) + 22|= 6
|−15 + 22|= 6
|−7|6= 6
-5 is not a solution because |−7|= 7, not 6.
2. Find the two solutions. Because there is a 2 being added to the left-side of the equation, we first need to subtract
it from both sides so the absolute value is by itself.
3
www.ck12.org
|6x − 11|+2 = 41
|6x − 11|= 39
.&
6x − 11 = 39
6x − 11 = −39
6x = 50
6x = −28
50
28
x=
or
x=−
6
6
25
1
14
2
=
or 8
= − or − 4
3
3
3
3
Check both solutions. It is easier to check solutions when they are improper fractions.
25
6
− 11 = 39
3
|50 − 11| = 39 X
6 − 14 − 11 = 39
3
and
|39| = 39
|−28 − 11|= 39 X
|−39|= 39
3. What is inside the absolute value is equal to 9 or -9.
1
x + 3 = 9
2
.&
1
1
x+3 = 9
x + 3 = −9
2
2
1
1
x = 6 or
x = −12
2
2
x = 12
x = −24
Test solutions:
1
(12) + 3 = 9
2
|6 + 3|= 9 X
|9|= 9
1
(−24) + 3 = 9
2
|−12 + 3|= 9 X
|−9|= 9
Vocabulary
Absolute Value
The positive distance from zero a given number is.
Problem Set
Determine if the following numbers are solutions to the given absolute value equations.
1. |x − 7|= 16; 9
4
www.ck12.org
Chapter 1. Solving Absolute Value Equations and Inequalities
2. | 14 x + 1|= 4; −8
3. |5x − 2|= 7; −1
Solve the following absolute value equations.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
|x + 3|= 8
|2x|= 9
|2x
3
+ 15|=
1
x − 5 = 2
3
x
+ 4 = 5
6
|7x
− 12|=
23
3
x + 2 = 11
5
|4x − 15|+1 = 18
|−3x + 20|= 35
Solve |12x − 18|= 0. What happens?
Challenge When would an absolute value equation have no solution? Give an example.
Absolute Value Inequalities
Objective
To solve absolute value inequalities.
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/60084
Khan Academy: Absolute Value Inequalities
Guidance
Like absolute value equations, absolute value inequalities also will have two answers. However, they will have a
range of answers, just like compound inequalities.
|x|> 1 This inequality will have two answers, when x is 1 and when −x is 1. But, what about the inequality sign?
The two possibilities would be:
5
www.ck12.org
Notice in the second inequality, we did not write x > −1. This is because what is inside the absolute value sign can
be positive or negative. Therefore, if x is negative, then −x > 1. It is a very important difference between the two
inequalities. Therefore, for the first solution, we leave the inequality sign the same and for the second solution we
need to change the sign of the answer AND flip the inequality sign.
Example A
Solve |x + 2|≤ 10.
Solution: There will be two solutions, one with the answer and sign unchanged and the other with the inequality
sign flipped and the answer with the opposite sign.
|x + 2|≤ 10
.&
x + 2 ≤ 10
x + 2 ≥ −10
x≤8
x ≥ −12
Test a solution, x = 0 :
|0 + 2| ≤ 10
|2| ≤ 10 X
When graphing this inequality, we have
Notice that this particular absolute value inequality has a solution that is an “and” inequality because the solution is
between two numbers.
If |ax + b|< c where a > 0 and c > 0, then −c < ax + b < c.
If |ax + b|≤ c where a > 0 and c > 0, then −c ≤ ax + b ≤ c.
If |ax + b|> c where a > 0 and c > 0, then ax + b < −c or ax + b > c.
If |ax + b|≥ c where a > 0 and c > 0, then ax + b ≤ −c or ax + b ≥ c.
If a < 0, we will have to divide by a negative and have to flip the inequality sign. This would change the end result.
If you are ever confused by the rules above, always test one or two solutions and graph it.
Example B
Solve and graph |4x − 3|> 9.
Solution: Break apart the absolute value inequality to find the two solutions.
|4x − 3|> 9
.&
4x − 3 > 9
4x > 12
x>3
6
4x − 3 < −9
4x < −6
3
x<−
2
www.ck12.org
Chapter 1. Solving Absolute Value Equations and Inequalities
Test a solution, x = 5 :
|4(5) − 3| > 9
|20 − 3| > 9 X
17 > 9
The graph is:
Example C
Solve |−2x + 5|< 11.
Solution: In this example, the rules above do not apply because a < 0. At first glance, this should become an “and”
inequality. But, because we will have to divide by a negative number, a, the answer will be in the form of an “or”
compound inequality. We can still solve it the same way we have solved the other examples.
|−2x + 5|< 11
.&
− 2x + 5 < 11
− 2x < 6
− 2x + 5 > −11
− 2x > −16
x > −3
x < −8
The solution is less than -8 or greater than -3.
The graph is:
When a < 0 for an absolute value inequality, it switches the results of the rules listed above.
Guided Practice
1. Is x = −4 a solution to |15 − 2x|> 9?
2
2. Solve and graph 3 x + 5 ≤ 17.
Answers
1. Plug in -4 for x to see if it works.
|15 − 2(−4)|> 9
|15 + 8|> 9
|23|> 9
23 > 9
7
www.ck12.org
Yes, -4 works, so it is a solution to this absolute value inequality.
2. Split apart the inequality to find the two answers.
2
x + 5 ≤ 17
3
.&
2
2
x + 5 ≤ 17
x + 5 ≥ −17
3
3
2
x ≤ 12
3
x ≤ 12 ·
2
x ≥ −22
3
3
2
x ≥ −22 ·
x ≤ 18
3
2
x ≥ −33
Test a solution, x = 0 :
2
(0) + 5 ≤ 17
3
|5| ≤ 17 X
5 ≤ 17
Problem Set
Determine if the following numbers are solutions to the given absolute value inequalities.
1. |x
− 9|> 4; 10
2. 12 x − 5 ≤ 1; 8
3. |5x + 14|≥ 29; −8
Solve and graph the following absolute value inequalities.
4.
5.
6.
7.
8.
9.
10.
11.
12.
|x + 6|> 12
|9 − x|≤ 16
|2x − 7|≥ 3
|8x
− 5|<
27
5
x + 1 > 6
6
|18
− 4x|≤
2
3
x − 8 > 13
4
|6 − 7x|≤ 34
|19 + 3x|≥ 46
Solve the following absolute value inequalities. a is greater than zero.
13. |x − a|> a
14. |x + a|≤ a
15. |a − x|≤ a
8