Solving Absolute Value Equations

Absolute value for most of us is a brief unit in math, and we hardly touch it again. However, if you need a refresher course or are diving into more complicated absolute value equations, allow us to help you!

To find absolute zero, count how far away a number is from zero on a number line. Both 6 and -6 are six units away from zero. You can represent this distance using the symbol │x│.  So, when you're writing absolute value for 6 and -6, it would look like this: │6│= 6 and │-6│ = 6.

Make it Opposite

Absolute value is never a negative number. It is always positive. The absolute value bars tell us that we do not want a negative number as a solution.

Let's take │6│= 6 again, but turn it to │-6│= x. What is the absolute value of -6? Already we know that it's six, but let's explore why.

If the number in the absolute value bars is greater than 0 or is 0, then you do not have to do anything to the solution. If the number in the absolute value bar is less than zero, then you use "-x" to flip the solution into a positive.

│-6│=  -x

But wait, we can't have a negative number as a solution. Multiply both sides by -x.

-(-6) = x

│6│= 6

Why Use Absolute Values

Perhaps the most common need for using absolute value is considering distance differences.  If two cars driving on an interstate highway are 30 miles apart, and one just passed mile marker 70, where is the other?  Since we don't know if the one at 70 is leading or not, the other is either at 40 or 100.

In reverse, if you know Car A is at mile marker 70 and Car B is at 40, easy math of A – B tells us they are 30 miles apart.  Unfortunately, subtracting A – B does not always work.  If someone tells you that B is at 70 and A is at 40, then A – B = -30.  Of course, the distance between them cannot be -30.

The absolute value function eliminates the need to know which is larger.

The symbolic relation should be │A – B│, or you can use │B - A│.  Both will give you a positive 30.

This is necessary for situations where you cannot use negative values, like square roots.  For example, √(B – A) in the first illustration will give you √(-30), which is not a real number.  To write it properly, you would use the absolute value:

√(│B – A│)

Absolute Values in Equations

Basic Equations

The most simple absolute value equation is one that asks you to solve for one number.

│x│ = 6.

What value can you substitute for x to get 6?  There are two solutions: x = -6 and x = 6.  You express this solution in brackets: {-6, 6}.

Now consider │x + 1│ = 6.

Again the operation inside the absolute value bars must equal -6 or 6.  To solve two separate equations.

x + 1 = 6                and        x + 1 = -6

x = 5                                    x = -7                     The solution set is {-7, 5}

A Little More Complication

│3x + 5│ = 6.   What’s inside the absolute value bars must equal -6 or 6. Break this up into two equations and solve.

3x + 5 = 6             and        3x + 5 = -6

x = 1/3                                 x = -11/3

The solution set is {-11/3, 1/3}

Check

│3(1/3) + 5│ = │1 + 5│ = 6 checks

and

│3(-11/3) + 5│ = │-11 + 5│ = 6  checks

Isolate the Absolute Value

4│2x - 3│ = 12

Here you must first isolate the absolute value by dividing both sides by 4.

│2x - 3│ = 3

Then set up two separate equations.

(2x – 3) = 3          and        (2x – 3) = -3

x = 3                                       x = 0

Check 

4│2(3) - 3│ = 4│3│ = 12   checks

and

4│2(0) - 3│ = 4│- 3│ = 12  checks

How Many Solutions?

For these examples the solution set had two values. However, consider these next two examples.

│x + 1│ = 0

Zero is neither positive nor negative. The only solution is x + 1 = 0 and x = -1. The solution set is {-1}.

│x + 1│ = -3

The left side has to be positive but the right side is negative. There is no number whose absolute value is -3. There is no solution. The solution set is { } or ꬾ (empty set).

Variable on Both Sides

│2x + 5│ = 11 – x

Set up two separate equations.  For the second equation, you must negate the entire left side.

2x + 5 = 11 – x                    and        2x + 5 = -(11 – x)

x = 2                                                   x = -16

 

Check 

│2(2) + 5│ =?  11 – (2)          and        │2(-16) + 5│ =?  -(11 – x)

│4 + 5│ =? 11 - 2                                │-32 + 5│ =?  -(11 – (-16))

│9│ = 9  checks

│-27│ = -(27)  Left side is positive, but the right side is negative.  It does not check.

Even though the algebra is right, one of the proposed solutions is not correct.  The solution set is {2}.

Extraneous Solutions

Absolute values force results of operations to be positive that otherwise may be negative. This means you must absolutely check your work. Sometimes your solution works; sometimes, one does but not the other; sometimes, neither works.

Summary

Solving absolute value equations is easy, but it can get tricky if you're faced with an equation that uses more than one value. Keep the following tips in mind:

1. Use algebra to isolate the absolute value, so it is the only thing on the right side of the equation.
2. Write two separate equations, the first being the original equation without the absolute value bars. For the second equation, change the signs on the right side to their opposite.
3. Solve each equation separately.
4. Check results by substituting values back into the original equation.  One or both of the values may be extraneous and should not be included in the solution.

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