Learning Objectives
Introduction
Tutorial
The absolute value measures the DISTANCE a number is away from the
origin (zero) on the number line. No matter if the number is
to the left (negative) or right (positive) of zero on the number line,
the DISTANCE it is away from zero is going to be positive. Hence, the absolute
value is always positive (or zero if you are taking the absolute value
of 0).
Example 1: What two numbers have an absolute value of 7?
Now, I want to explain the thought behind it because this is going to help us to understand how to solve absolute value equations. I really want to emphasize the fact that there are two numbers that are the same distance away from the origin, the positive number and its opposite. The thought behind this is there are two places on the number line that are 7 units away from zero - both 7 and -7.
Solving an Absolute Value Equation
x = d OR x = -d
(two equations are set up) If d is NEGATIVE and |x| = d, then
No solution
This is because distance (d)
can not be negative.
If you need a review on solving linear equations, feel free to go to Tutorial 14: Linear Equations in One Variable.
If you need a review on solving quadratic equations, feel free to go
to Tutorial 17: Quadratic Equations.
Example 2: Solve the absolute value equation .
Using the definition of absolute value, there will be two equations
that we will need to set up to get rid of the absolute value because there
are two places on the number line that are 7 units away from zero: 7 and
-7.
*Inv. of add. 5 is sub. 5
*Inv. of mult. by -3 is div. by -3
*Inv. of add. 5 is sub. 5
*Inv. of mult. by -3 is div. by -3
If you would have only come up with an answer of x = -2/3, you would not have gotten all solutions to this problem.
Again, it is important to note that we are using the definition of absolute value to set the two equations up. Once you apply the definition and set it up without the absolute value, you just solve the linear equation as shown in Tutorial 14: Linear Equations in One Variable.
There are two solutions to this absolute value equation: -2/3
and 4.
Example 3: Solve the absolute value equation .
Using the definition of absolute value, there will be two equations
that we will need to set up to get rid of the absolute value because there
are two places on the number line that are 3 units away from zero: 3 and
-3.
*Quad. eq. in standard form
*Factor
the trinomial
*Use Zero-Product
Principle
*Set 1st factor = 0 and solve
*Set 2nd factor = 0 and solve
*Quad. eq. in standard form
*Factor
the trinomial
*Use Zero-Product
Principle
*Set 1st factor = 0 and solve
*Set 2nd factor = 0 and solve
If you would have only come up with an answer of x = -4 and 3, you would not have gotten all solutions to this problem.
Again, it is important to note that we are using the definition of absolute value to set the two equations up. Once you apply the definition and set it up without the absolute value, you just solve the quadratic equation as shown in Tutorial 17: Quadratic Equations.
There is are four solutions to this absolute value equation: -4,
3, -3, and 2.
Example 4: Solve the absolute value equation .
Answer: No solution.
Example 5: Solve the absolute value equation .
Using the definition of absolute value, there will be only one equation
that we will need to set up to get rid of the absolute value, because there
is only one place on the number line that is 0 units away from zero: 0.
First equation:
*Inv. of sub. 4 is add 4
*Inv. of mult. by 2 is div. by 2
Again, it is important to note that we are using the definition of absolute value to set the equation up. Once you apply the definition and set it up without the absolute value, you just solve the linear equation as shown in Tutorial 14: Linear Equations in One Variable.
There is only one solution to this absolute value equation: 2.
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1c: Solve each absolute value equation.
Need Extra Help on these Topics?
The following is a webpage that can assist you in the topics that were covered on this page:
http://www.sosmath.com/algebra/solve/solve0/solve0.html#absolute
Only look at problem 1. Problem 1 at this webpage helps you with solving absolute value equations.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
WTAMU > Virtual Math Lab > College Algebra
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 16, 2009 by Kim Seward.
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