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Solving Absolute Value Equations

The absolute value of any number is the distance between the number and zero on the number line. Absolute value is also referred to as modulus (or mod in short). Absolute value of a variable is defined as its non-negative value. When we find absolute value, we ignore the sign. We can write that
$\left | x \right |=x\ and\ \left | -x \right |=x$$\left | x \right |$ is pronounced as "absolute value of x" or "mod of x".
An equation which contains absolute values in it is known as absolute value equation. In order to solve an absolute value equation, we use the following rule:
Solving Absolute Value EquationsFollow the steps given below while solving absolute-value equations.
Step 1: Apply above rule to the given equation with absolute value and separate it in two equations.
Step 2: Solve each equation separately.
Step 3: After solving, cross check your answers to determine the authenticity of each answer.
Step 4: Sometimes, all the values do not satisfy the equation. Ignore the one which does not satisfy given equation. The remaining value will be the required solution.

Let us consider an example to understand this concept.
Example: $\left | 2x+5 \right |=9$
Solution: $\left | 2x+5 \right |=9$
Using above rule:
2x + 5 = 9 or 2x + 5 = -9

Now, solve both of them,
2x + 5 = 9
2x = 9 - 5
2x = 4
x = 2

2x + 5 = -9
2x = - 9 - 5
2x = -14
x = -7

Now, check the answers:
Substituting x = 2 in the given equation, we get
$\left | 4+5 \right |$
= $\left | 9 \right |$
= 9
Hence, x = 2 is a solution.

Substituting x = -7 in the given equation, we get
$\left | -14+5 \right |$
= $\left | -9 \right |$
= 9
Hence x = -7 is a solution.

So, x = 2, -7

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