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,