This document provides an overview of solving absolute value equations through examples of basic, intermediate, and advanced problems. It begins with definitions of absolute value and examples of basic absolute value equations. Intermediate examples show how to isolate the absolute value term and then solve the resulting simple equations. More complex advanced examples involve fractions, which are solved by clearing fractions to get a common denominator before isolating and solving the absolute value term. The goal is to build skill in solving various types of absolute value equations.

1. A Silent Lesson On AbsoluteValue

2.  Click on the options below in order to move
through the lab.
Definition of AbsoluteValue
Basic AbsoluteValue Problems
Intermediate AbsoluteValue Problems
Advanced AbsoluteValue Problems

3.  Definition (Lay-man’s)
Absolute value is a mathematical object which
re-expresses a number as its distance from
zero. (De-noted by the brackets)
 Examples:
9.1 = 9.1 −3.8 =3.8
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4.  We would look at an equation like
2𝑥 − 1 = 7 and read “the absolute value of
2𝑥 − 1 is 7.”
 Meaning that 2𝑥 − 1 lives seven units away
from zero on the number line.
2𝑥 − 1 = 7 2𝑥 − 1 = 72𝑥 − 1 = −7
There are two solutions because there are two
places on the number line that are seven units
away from zero! Namely, 7 and -7!
Home Solving Basic Equations

5.  Ex: Solve the following for 𝑥.
𝑥 − 6 = 4
Let’s start by
thinking about the
definition…
What I’m saying here, is that
𝑥 − 6 is four units away from
zero on the number line.
𝑥 − 6 = 4𝑥 − 6 = −4
Now I have two simple equations set up, so I solve those for x!
𝑥 = 2 𝑥 = 10
𝑥 ∈ {2,10}
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To “model” these solutions, we typically use points on the number line.
Next Example:

6.  Now you try!
 Ex: Solve the following for 𝑥.
3𝑥 + 2 = 8
Click for solutions…
𝑥 ∈ −
10
3
, 2
Home
Click here to see video of solution!
Intermediate Problems

7. Home Intermediate Problems

8.  Ex: Solve the following for 𝑥.
4 − 3 2𝑥 − 3 = −5
We need to get the
absolute value portion of
the equation by itself, then
it’s a simple problem!
Ultimately, we want to get
rid of the 4and the -3.
The trick is just getting the
order right!
Since the -3 is multiplying, let’s get rid
of that last. So we’ll get rid of the 4
first! We do this be subtracting.
−4 −4
−3 2𝑥 − 3 = −9
Now, we want to get rid of the -3 that
is multiplying, so we divide both sides
by -3.
−3 −3
2𝑥 − 3 = 3
2𝑥 − 3 = 32𝑥 − 3 = −3
𝑥 = 3𝑥 = 0
𝑥 ∈ {3, −}0Home
Next Example

9.  Now you try!
 Ex: Solve the following for 𝑥.
2 4𝑥 + 2 − 1 = 11
Click for solutions…
𝑥 ∈ −2,1
Click here to see video of solution!
Home Advanced Problems

10. Home Advanced Problems

11.  Truth be told, these questions aren’t any
different from the intermediate questions
there’s just fractions involved.
Home

12.  Ex: Solve the following for 𝑥.
2
2
3
𝑥 −
1
5
+ 1 = 13
Don’t freak out at the fractions! Just
do what you know how to do first!
Get the absolute value expression by
itself first!
−1 +1
2
2
3
𝑥 −
1
5
= 12
−1
2 2
2
3
𝑥 −
1
5
= 6
Let’s get rid of all of
the fractions at once!
We do this by multiplying both sides of
the equations by the smallest number
three an five (the denominators) have in
common.
2
3
𝑥 −
1
5
= −6
2
3
𝑥 −
1
5
= 6
Since 3, and 5 have nothing in common we multiply both equations by 15!
15( ) 15( )( )15 ( )15
30
3
𝑥 −
15
5
= −90
30
3
𝑥 −
15
5
= 90
10𝑥 − 3 = 9010𝑥 − 3 = −90
𝑥 = −
87
10
𝑥 =
93
10
𝑥 ∈ −
87
10
,
93
10
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13.  Now you try!
 Ex: Solve the following for 𝑥.
2 + 3
1
5
𝑥 −
1
3
= 11
Click for solutions…
𝑥 ∈ −
40
3
,
50
3
Click here to see video of solution!
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14. Home

15.  You’ve just finished the lab! Hopefully you’ve
built sufficient skill to start solving some
really challenging problems.
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