Inequalities and Absolute Value

There are a couple of ways to think of taking the absolute value of a number. The different ways of thinking about it helps sometimes when proving. The first way of thinking of absolute value involves squaring and taking the square root.


The squaring makes all negative numbers positive and taking the square root takes us back to the original number. Technically, taking the square root of a positive number gives us two answers, the original number along with its negative. So when we take the square root of a number involving absolute value we take the positive root. The second way involves the sign of x:

Let's try working a problem with an absolute value. Consider the general inequality:

|x - c| < .

Using the definition of absolute value we come up with two inequalities:

x - c < and - (x - c) < .

Solving for x we get:
x < c + and x > c -

put together
c - < x < c + .

What this is saying is that x stands for all numbers units within c.

There is a special inequality that has a name. It comes up in linear algebra as well. It is called the triangle inequality:

|a + b| |a| + |b|.


The last step of the proof uses the definition that |a| = a if a > 0. Since |a|0 and |b|0:

Let's try solving a more specific inequality with an absolute value:

|x - 5| < 10.

We use our definition of absolute value to give us two inequalities:

x - 5 < 10 and -(x - 5) < 10.

Remembering that multiplying an inequality by -1 reverses it, the two simplify to:
x < 15 and x > -5.

The solution is all the numbers in between which in interval notation looks like (-5,15). This is not to be confused with x-y coordinates even though the two look the same.