Absolute value means magnitude of a variable without the sign of it. Suppose you walk in the positive x-axis 3 units starting from 0, then x = 3. Now if you walk in the negative x-axis 3 units, mathematically x = -3 but the distance that you travelled in either direction is the same and equal to 3 units.

Absolute values are represented by | x | and read as magnitude or absolute value of x.

If x is positive, |x| = x         Ex.  |7| = 7

If x is negative, |x| = -x       Ex. |-7| = -(-7) = 7

Example: Solve the following equation

| x - 10 | = 3

If (x-10) is positive, we can write,

x - 10 = 3 \\

or, x = 13

If (x-10) is negative, we can write,

-(x-10) = 3

or, -x + 10 = 3 \\

or, x = 7

We  need to verify if the solution satisfies the given equation. By substituting 13 and 7 in the above equation we get

| 13 - 10 | = 3 \quad \quad \quad | 7 - 10 | = 3

or, | 3 | = 3 \quad \quad \quad or, | -3 | = 3

Since both the values of x satisfy the given equation, solution is 13, 7. This may not always true as we will see in the next example

Example: Solve the following equation

| x | + | 2x - 17 | = 13

In this example there are two magnitude terms, so we need to consider all possible cases.

Case 1: When x is positive and (2x-17) is positive

x + 2x - 17 = 13

or, 3x = 30

or, x = 10

Case 2: When x is positive and (2x-17) is negative

x - (2x-17) = 13 \\

or, x - 2x + 17 = 13 \\

or, -x = -4 \\

or, x = 4 \\

Case 3: When x is negative and (2x-17) is positive

-x + 2x - 17 = 13 \\

or, x = 30 \\

Case 4: When x is negative and (2x-17) is negative

-x - 2x +17 = 13 \\

or, -3x = -4 \\

or, x = \frac{4}{3}

Now let’s substitute the values of x in the given equation to see which solution satisfies the equation

For x =10,  |10| + |2×10 – 17| = 13           

or,  |10| + |3| = 13

or, 13 = 13

For x = 4, |4| + |2×4 – 17|= 13

or,  |4| + |-9| = 13

or, 13 = 13

For x = 30, |30| + |2×30 -17| = 13

or,  |30| + |43| = 13

or, 73 \neq 13

For x= \frac{4}{3}, |\frac{4}{3}| + |2×\frac{4}{3} – 17| = 13

or, |\frac{4}{3}| + |\frac{8}{3}-17| = 13       

or, |\frac{4}{3}| + |-\frac{43}{3}| = 13          

or, |\frac{47}{3}| \neq 13

So, the solution is 10, 4