Equations with Absolute Values - Examples, Exercises and Solutions

Question Types:
Absolute value: Solve absolute value equations with a number outside the absolute value barsEquations with Absolute Values: Quadratic equationAbsolute value: Identifying Number oppositesEquations with Absolute Values: Absolute value on both sides of the equationEquations with Absolute Values: Double absolute valueAbsolute value: Solving the Absolute valueEquations with Absolute Values: Identifying functions from graphsEquations with Absolute Values: System of equationsEquations with Absolute Values: Solving the problem

The "absolute value" may seem complicated to us, but it is simply the distance between a given number and the figure 0 0 . 

What is absolute value?

An absolute value is denoted by ││ and expresses the distance from zero points.
The absolute value of a positive number - will always be the number itself.
For example: │2│=2│2│= 2
Absolute value of a negative number: will always be the same number, but positive.
For example: │−3│=3│-3│=3
Note that the absolute value of a number will always be a positive number since distance is always positive.

Un valor absoluto se denota por ││

The absolute value of a number is the distance between it and the number 0.

For example:

  • The distance between the number +7 +7 and 0 0 is 7 7 units. Therefore, the absolute value of +7 +7 is 7 7 .
  • The distance between the number −7 -7 and 0 0 is also 7 7 units. Therefore, the absolute value of −7 -7 will also be 7 7 . 

As we can see, from the point of view of absolute value, it doesn't matter if the number is positive or negative.

To denote the absolute value, the number is written between two vertical lines.

Detailed explanation

Suggested Topics to Practice in Advance

  1. Numerical Value

Practice Equations with Absolute Values

Question 1

\( \left|18\right|= \)

Question 2

\( \left|-2\right|= \)

Question 3

\( \left|6x-12\right|=6 \)

Question 4

\( \left|0.8\right|= \)

Question 5

\( \left|3\right|= \)

Examples with solutions for Equations with Absolute Values

Exercise #1

∣18∣= \left|18\right|=

Video Solution

Step-by-Step Solution

The "absolute value" can be viewed as the distance of a number from 0.
Therefore, the absolute value will not change the sign from negative to positive, it will always be positive.

Answer

18 18

Exercise #2

∣−2∣= \left|-2\right|=

Video Solution

Step-by-Step Solution

When we have an exercise with these symbols || we understand that it refers to absolute value.

Absolute value does not relate to whether a number is positive or negative, but rather checks how far it is from zero.

In other words, 2 is 2 units away from zero, and -2 is also 2 units away from zero,

Therefore, absolute value essentially "zeroes out" the negativity of the number.

 

|-2| = 2

 

Answer

2 2

Exercise #3

∣6x−12∣=6 \left|6x-12\right|=6

Video Solution

Step-by-Step Solution

To solve this exercise, we need to note that the left side is in absolute value.

Absolute value checks the distance of a number from zero, meaning its solution is always positive.

 

Therefore, we have two possibilities, either the numbers inside will be positive or negative,

In other words, we check two options, in one what's inside the absolute value is positive and in the second it's negative.

 

6x-12=6

6x=18

x=3

This is the first solution

 

-(6x-12)=6
-6x+12=6
-6x=6-12
-6x=-6
6x=6
x=1

And this is the second solution,

So we found two solutions,

x=1, x=3

And that's the solution!

Answer

x=1 x=1 , x=3 x=3

Exercise #4

∣0.8∣= \left|0.8\right|=

Video Solution

Answer

0.8 0.8

Exercise #5

∣3∣= \left|3\right|=

Video Solution

Answer

3 3

Question 1

\( −\left|-18\right|= \)

Question 2

\( \left|-19\frac{1}{4}\right|= \)

Question 3

\( \left|3^2\right|= \)

Question 4

\( \left|x-10\right|=0 \)

Question 5

\( \left|x+1\right|=5 \)

Exercise #6

−∣−18∣= −\left|-18\right|=

Video Solution

Answer

−18 -18

Exercise #7

∣−1914∣= \left|-19\frac{1}{4}\right|=

Video Solution

Answer

1914 19\frac{1}{4}

Exercise #8

∣32∣= \left|3^2\right|=

Video Solution

Answer

9 9

Exercise #9

∣x−10∣=0 \left|x-10\right|=0

Video Solution

Answer

x=10 x=10

Exercise #10

∣x+1∣=5 \left|x+1\right|=5

Video Solution

Answer

Answers a + b

Question 1

\( \left|x-1\right|=6 \)

Question 2

\( \left|x\right|=5 \)

Question 3

\( \left|x\right|= \)

Question 4

\( -\lvert4^2\rvert= \)

Question 5

Are the numbers opposite?

\( \left|-3\right|,\left|3\right| \)

Exercise #11

∣x−1∣=6 \left|x-1\right|=6

Video Solution

Answer

x=−5 x=-5 , x=7 x=7

Exercise #12

∣x∣=5 \left|x\right|=5

Video Solution

Answer

Answers a + b

Exercise #13

∣x∣= \left|x\right|=

Video Solution

Answer

x x

Exercise #14

−∣42∣= -\lvert4^2\rvert=

Video Solution

Answer

−16 -16

Exercise #15

Are the numbers opposite?

∣−3∣,∣3∣ \left|-3\right|,\left|3\right|

Video Solution

Answer

No

Topics learned in later sections

  1. Absolute Value Inequalities
  2. Inequalities with Absolute Value