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Absolute Value Equations

Absolute Value Equations

Author: Sophia Tutorial
Description:

Solve an absolute value equation.

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Tutorial
what's covered
  1. Absolute Value Review
  2. Equations in the form lxl
  3. Equations in the form lmx + bl

1. Absolute Value Review

The absolute value of a number is the distance from zero on the number line. Distance is never a negative value, so absolute value never returns a negative.

We can look at the number line and see that the absolute value of a positive number and the absolute value of a negative number are both positive:

File:5595-av1.png

Both –4 and 4 are 4 units away from zero on the number line. We can say that open vertical bar 4 close vertical bar equals 4 and open vertical bar negative 4 close vertical bar equals 4.

This leads to a piecewise definition of absolute value:

formula
Absolute Value
open vertical bar x close vertical bar equals open curly brackets table attributes columnalign left end attributes row cell x comma space w h e n space x greater or equal than 0 end cell row cell negative x comma space w h e n space x less than 0 end cell end table close

This means that if x is zero or greater, the value of x does not change when we apply the absolute value. However, if x is negative, we change the sign of x when applying the absolute value, so as to make it positive.


2. Equations in the Form |x|

When we solve any absolute value equation, we just remember that the expression can be positive or negative, and still have the same absolute value. For this reason, we create two separate equations: one with a positive value, and one with a negative value, for the expression inside absolute value bars:

8 equals open vertical bar x close vertical bar

8 equals x comma space space space space space space space space space short dash 8 equals x
Create two equations
x equals short dash 8 comma space 8
Our solution

To solve absolute value equations, we can remove the absolute value bars only if we create two equations: one equation will look nearly identical to the original absolute value equation, while the second equation will consider the case when the expression is negative.

Next, we will consider more complex absolute value equations


3. Equations in the Form |mx + b|

Once again, we are going to solve this equation by creating two separate equations without absolute value bars. One equation will contain the expression exactly as it appears within the absolute value bars. The second equation must consider the case when the expression has the opposite value. That is, we reverse the sign on the other side of the equation:

14 equals open vertical bar 3 x plus 2 close vertical bar

14 equals 3 x plus 2 comma space space space space space space short dash 14 equals 3 x plus 2
Create two equations

Next, we solve each equation individually, and include both solutions for x as solutions to the absolute value equation.

14 equals 3 x plus 2
First equation
12 equals 3 x
Subtract 2
4 equals x
Divide by 3
short dash 14 equals 3 x plus 2
Second equation
short dash 16 equals 3 x
Subtract 2
fraction numerator short dash 16 over denominator 3 end fraction equals x

Divide by 3
x equals short dash 5.33 with bar on top comma space 4
Our Solution

big idea
Create two equations, each without absolute value bars. One equation will remain the same in every other respect, while the second equation will have the opposite value (reversed sign) on the other side of the equation.

summary
The absolute value of a number is its distance from 0 on the number line, and is always non-negative. We discussed two types of absolute value equations: equations in the form lxl and equations in the form lmx + bl. When solving absolute value equations, we must consider both the positive and negative values of the expression inside the absolute value bars, because they can equal the same value once we take the absolute value. Also, when solving absolute value equations, the first step is to isolate the absolute value expression on one side of the equation.

Formulas to Know
Absolute Value

open vertical bar x close vertical bar equals open curly brackets table attributes columnalign left end attributes row cell x comma space w h e n space x greater or equal than 0 end cell row cell negative x comma space w h e n space x less than 0 end cell end table close