**Basic Inequalities**

Inequalities are solved similar to equations, with rules of inequality. For example, the solving of 3x+4<7 is as follows:

This can be graphed on a number line with a dot where the 1 is. For a greater-than or less-than inequality, an open dot(o) is used to represent the number. For a greater-than-or-equal-to or less-than-or-equal-to inequality, use a closed dot(•).

This would be graphed:

< ——-

1

Note the open dot over 1.

In an inequality, one major difference is that when dividing or multiplying by a negative number, the inequality sign is switched.

Notice the flipped sign. When multiplying both sides by -3, the ≤ becomes ≥.

This would be graphed as:

———->

9

**Compound Inequalities**

Inequalities can be combined with *and* or *or*.

An *and* inequality has multiple inequality signs. For example:

1 < x < 10

Inequality operations can still be performed, but must affect all sections. When dividing or multiplying by a negative number, you must flip the signs.

*And* inequalities can also be graphed, by connecting two dots. For example, 1 < x < 10 is graphed:

O——O

1 10

*Or* inequality:

An *or* inequality is two inequalities joined by an *or*

Which is graphed as two separate inequalities.

**Interval Notation**

Interval notation is a way of describing inequalities as a set of numbers from *a* to *b*.

For example, x>3 would be described as:

Because it starts at 3 and goes to infinity.

A parenthesis is used for both sides because it approaches but never reaches 3 and infinity. If the inequality was x≤5 the interval notation would be:

Now a square bracket is by 5 because *x* can reach 5.

For or inequalities, two interval notation groups are used, separated by *or*.