SAT/ACT Math: Absolute Value

Absolute value is among the most challenging quant topics on the SAT and the ACT, most probably, because such questions dont feature much as a part of the school syllabus. 

Absolute values are expressions within parentheses, which have the superpower to turn every expression into a non-negative number. You must follow all the PEMDAS/BODMAS rules before the number gets out of the parentheses. For example: |3 - 10*2| = |3 - 20| = |-17| = 17.

Noticed the superpower of the parentheses here? They converted a negative number to a positive number. So, if |x| = 2, then x can be both a positive or a negative 2. 

Why does absolute value have to be a non-negative integer? It is because absolute value means the distance between the number and the origin on a number line and, as you know, the distance between any two things cannot be negative.

So, keep in mind:

• If the number inside the parentheses is positive, it stays positive.
• If the number inside the parentheses is 0, then it stays 0.
• If the number inside the parentheses is negative, then it turns positive.

Now, let's apply this understanding to solve absolute value questions.

Scenario 1:

The quantity inside the parentheses is non-negative. In this case, we simply remove the parentheses.

So, if x - a ≥ 0 in |x - a|, then |x - a| = x - a.

Scenario 2:
The quantity inside the parentheses is negative. In this case, we remove the parentheses and multiply the expression with a negative sign.

So, if x - a < 0 in |x - a|, then |x - a| = -(x - a) = a - x.

Example 1:

Solve for x if |x - 1| = 3.

There are two possibilities:

a) x - 1 ≥ 0

So, x - 1 = 3,

x = 3 + 1,

x = 4

b) x - 1 < 0

So, -(x - 1) = 3,

-x + 1 = 3,

-x = 3 - 1,

-x = 2,

x = -2

So, there are two possible values for x: 4 and -2.

However, you should verify both the values by placing them in the original equation. 

Applying x = 4 to |x - 1| = 3,

|4 - 1| = 3.

It works.

Applying x = -2 to |x - 1| = 3,

|-2 - 1| = 3,
|-2 - 1| = |-3| = 3.

This works too.

Hence, there are two possible values for x.

Example 2:

Solve for x, if |x - 3| = 2x.

There are two possibilities:

a) x - 3 ≥ 0,

So, x - 3 = 2x,

x - 2x = 3,

-x = 3,

x = -3

b) If x - 3 < 0,

So, -(x - 3) = 2x,

-x + 3 = 2x,

-x - 2x = -3,

-3x = -3,

x = 1

So, there are two possible values for x: -2 and 4.

However, you should verify both the values by placing them in the original equation. 

Applying x = -3 to |x - 3| = 2x,

|-3 - 1| = 2(-3).
|-4| = -6. 

It doesn’t work.

Applying x = 1 to |x - 3| = 2x,

|1 - 3| = 2(1).
|-2| = 2.

This works.

Hence, there is only possible value of x in this case.

We hope you have gained a clearer understanding of this concept.
Happy prepping! Best of luck for your test!