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Solving absolute value equations


When solving absolute value equations, many people routinely solve these without really understand deep concepts involving absolute value. It is the goal of this lesson to remedy this common pitfall.


Absolute value definition:

If x is positive, | x | = x

If x is negative, | x | = -x

Example #1:

Solve for x when | x | = 4

After applying the definition to example #1, you will have two equations to solve

In fact, when solving absolute value equations, you will usually get two solutions. That is important to keep in mind

If x is positive, | x | = x, so the first equation to solve is x = 4. Done because x is automatically isolated

If x is negative, | x | = -x, so the second equation to solve is -x = 4.

You can write -x = 4 as -1x = 4 and divide both sides by -1 to isolate x.

(-1/-1)x = 4/-1

1x = -4

x = -4

Therefore, the solutions are 4 and -4

Example #2:

Solve for x when | x − 5 | = 2

Before, we apply the definition, let's make a useful substitution

Let y = x − 5, so | x − 5 | = 2 becomes | y | = 2. You must understand this step. No excuses!

Now, let's apply the definition to | y | = 2. Again, you will have two equations to solve

Once again, when solving absolute value equations, you will usually get two solutions.

If y is positive, | y | = y, so the first equation to solve is y = 2. No, you are not done! You have to substitute x − 5 for y

After substitution, y = 2 becomes x − 5 = 2

x − 5 = 2

x + -5 = 2

x + -5 + 5 = 2 + 5

x = 7

If y is negative, | y | = -y, so the second equation to solve is -y = 2.

You have to substitute x − 5 for y

You get -( x − 5) = 2. Notice the inclusion of parenthesis this time

-(x − 5) = 2

-(x + -5) = 2

-x + 5 = 2

-x + 5 − 5 = 2 − 5

-x = 2 + -5

-x = -3

-1x = -3

(-1/-1)x = -3/-1

x = 3

The solutions are 3 and 7

Example #3:

Solve for x when | 3x + 3 | = 15

Before, we apply the definition, let's again make a useful substitution

Let y = 3x + 3, so | 3x + 3 | = 15 becomes | y | = 15.

Now, let's apply the definition to | y | = 15. Again, you will have two equations to solve.

If y is positive, | y | = y, so the first equation to solve is y = 15. You have to substitute 3x + 3 for y

After substitution, y = 15 becomes 3x + 3 = 15

3x + 3 = 15

3x + 3 − 3 = 15 − 3

3x = 12

(3/3)x = 12/3

x = 4

If y is negative, | y | = -y, so the second equation to solve is -y = 15.

You have to substitute 3x + 3 for y

You get -( 3x + 3) = 15. Notice the inclusion of parenthesis this time

-3x + -3 = 15

-3x + -3 + 3 = 15 + 3

-3x = 18

(-3/-3)x = 18/-3

x = -6 The solutions are -6 and 4

Solving absolute value equations should be straightforward if you follow my guidelines above


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