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.

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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|.

Proof

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