Simple Inequalities

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In Mathematics, an inequality is a statement about the relative size or order of two objects.

The notation $a < b \!\$ means that a is less than b and
The notation $a > b \!\$ means that a is greater than b.

Therefore a is not equal to b. These relations are known as strict inequality; in contrast

$a \le b$ means that a is less than or equal to b;
$a \ge b$ means that a is greater than or equal to b;
$a \not= b$ means that a is not equal to b and
$a = b$ means that a is equal to b.

An additional use of the notation is to show that one quantity is much greater than another, normally by several orders of magnitude.

The notation $a \gg b$ means that a is much greater than b.
The notation $a \ll b$ means that a is much less than b.

If the sense of the inequality is the same for all values of the variables for which its members are defined, then the inequality is called an "absolute" or "unconditional" inequality. If the sense of an inequality holds only for certain values of the variables involved, but is reversed or destroyed for other values of the variables, it is called a conditional inequality. The sense of an inequality is not changed if both sides are increased or decreased by the same number, or if both sides are multiplied or divided by a positive number; the sense of an inequality is reversed if both members are multiplied or divided by a negative number.

Inequalities

Now that you have a basic understanding of what the symbols for inequalities mean, let's look at how they apply to this course.

Typically, we are going to look for a special classification of numbers. We want to limit our search for answers to specific locations, or domains. You'll use a few tools to help you graph.

This also applies to sets. Sets are specific values that we can use to find answers. Inequalities do the same thing. They limit the number of possible answers.

Let's discuss graphing inequalities!

Graphing Simple Inequalities

We have different types of inequalities, as discussed above. We have the inclusive types (as in ≥ or ≤) and we have the non-inclusive types (> or <). We'll take a normal number line line the one shown here:

and we'll use it to graphically represent our set or domain.

Inclusive means that the inequality includes the number we're starting at. The inequality x ≤ 3 means that we are looking at all numbers less than 3 as well as 3 itself (note the equal sign under the less than sign). This is why this type is inclusive. It includes the number.

Inclusive inequalities are graphed with closed dots or circles. The graph of x ≤ 3 looks like this: