# Absolute Value and Inequalities

Examples.

The minus signs don't cancel; they're "blocked" by the absolute value.

Geometrically, the absolute value of a number is its distance from the origin 0. So -13.8 is units from 0.

More generally, is the distance from foo to bar. For example,

Example. Is the following equation true?

The equation is true:

So .

Example. Is the following equation true?

The equation is not true, because

So .

Example. Is the following algebraic operation legal (for all a and b)?

In the last two examples, I saw one case where it works and another where it doesn't. Hence, the operation is not legal for all a and b (and in particular, it doesn't count as a rule of algebra).

Example. Is the following equation true?

The operation is valid:

The two sides are equal.

Example. Simplify .

In simplifying algebraic expressions, it's often helpful to work "from the inside out". First, I change to 3 and to 4:

Next, I do and :

Finally, :

That is,

Example. Simplify .

First, , , and :

(Notice again how I'm evaluating the expression "inside out".)

Next, and :

The rest is just arithmetic:

Example. Graph the inequality .

Example. Graph the inequality .

Example. What inequality is represented by the following picture?

. The open (white) circle means the point -4 is {\it not} included; the closed (black) circle means the point 2 is included.

Example. What inequality is represented by the following picture?

Here is an easy way to write down inequalities from number line pictures. First, put 's on either side of each number. Be sure they're 's, not 's (i.e. be sure they all point the same way).

Now put an x under each shaded part, and pick off the inequalities corresponding to the shaded parts: or .

But note that you cannot write this as " ", because (ignoring the x), this says " ", which is {\it false}.

Example. Which is bigger, or ?

There are several reasonable ways to figure this out. One approach is to compute the decimal values of these fractions. Since while , it follows that .

You can compute the decimal values with a calculator, but also by hand, using long division.

Example. Which is bigger, or ?

Since while , it follows that .

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