Solving Absolute Value Equations

As we know, the absolute value of a number refers to its distance from 0 on the number line. It is represented as |x|, which defines the magnitude of any integer "x". The absolute value of any value, whether positive or negative, real or complex, will be a non negative real number, regardless of which sign it has. The operation is represented by two vertical lines, |x|, which we read as the absolute value of x.

For example, 8 is the absolute value for both +8 and -8.

$\left|-8\right|=8$ and $\left|\mathrm{+8}\right|=8$

The absolute value of a number, whether the number is positive or negative, is always a non-negative number. The absolute value of 0 is 0.

Examples of absolute values

$\left|-9\right|=9$

$\left|17\right|=17$

$\left|0\right|=0$

$\left|-27\right|=27$

$\left|7-4\right|=\left|3\right|=3$

$\left|12-24\right|=|-12|=12$

$\left|-4\times 4\right|=|-16|=16$

Properties of absolute values

There are several properties of absolute values that hold true in all equations. They are as follows:

• Non-negativity: $\left|x\right|$ is always greater than or equal to 0
• Multiplicative: $|x\times y|=\left|x\right|\times \left|y\right|$
• Subadditivity: $|x+y|$ is always less than or equal to $\left|x\right|+\left|y\right|$

Solving absolute value equations

Absolute value equations are equations that have one or more variables in the absolute value bars.

When solving absolute value equations, there are two cases to consider:

Case 1: The expression inside the absolute value bars is positive.

Case 2: The expression inside the absolute value bars is negative.

Example 1

Take the expression $\left|4x+2\right|=18$ as an example.

For this to be true, either

$4x+2=18$ or $4x+2=-18$

You need to solve both equations to get the correct answers to the absolute value equation.

Case 1:

$4x+2=18$

$4x=16$

$x=\frac{16}{4}=4$

Case 2:

$4x+2=-18$

$4x=-20$

$x=-\frac{20}{4}=-5$

So there are two solutions to the absolute value equation $\left|4x+2\right|=18,x=4,x=-5$ .

Example 2

It's also possible for an absolute value equation to have only one solution.

For example, the single solution to $\left|x+3\right|=0$ because are cases break down into

Case 1:

$\left|x+3\right|=-0$

Case 2:

$\left|x+3\right|=\mathrm{+0}$

which both have the same solution, $x=-3$

Example 3

It is also possible for an absolute value equation to have no solutions.

$\left|5x+1\right|=-6$

Since the absolute value of any expression is positive, there is no value of x for which this is true.

Practice working with absolute value equations

a. Arrange the given numbers in ascending order.

$\left|18\right|,-\left|-7\right|,\left|17\right|,\left|93\right|,|-68|,\left|6\right|,|-7|$

First, find the absolute value of each number:

$18,-7,17,93,68,6,7$

Then arrange the numbers in order from smallest to largest:

$-7,6,7,17,18,68,93$

b. Solve the equation $|x+4|=1$ .

First, write the two equations from the given equation in such a way that

$x+4=1$

$x+4=-1$

Solve the above equations to get the possible values of x.

$x+4=1$

$x=1-4$

$x=-3$

$x+4=-1$

$x=-1-4$

$x=-5$

So the two possible values of x are -3 and -5.

c. Solve the equation $\left|2x-5\right|=9$

First, write the two equations from the given equation in such a way that

$2x-5=9$

$2x-5=-9$

Solve the above equations to get the possible values of x.

Case 1:

$2x-5=9$

$2x=9+5$

$2x=14$

$x=\frac{14}{2}$

$x=7$

Case 2:

$2x-5=-9$

$2x=-9+5$

$2x=-4$

$2x=-\frac{4}{2}$

$x=-2$

So the two possible values of x are 7 and -2.

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Get help learning about solving absolute value equations

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