Introduction to Inequalities

Learning Objectives

Understand the meaning of the four inequality symbols: <, >, ≤, ≥.

Solve simple one-step inequalities.

Represent the solution to an inequality on a number line.

Explain the rule for multiplying or dividing an inequality by a negative number.

More Than Just Equal

In math, we often want to compare quantities that are not equal. We use inequalities to do this. An inequality is a mathematical statement that compares two values, showing that one is less than, greater than, or not equal to another.

The Four Inequality Symbols

There are four main symbols used in inequalities:

< : Less than (e.g., 3 < 5 means '3 is less than 5')

> : Greater than (e.g., 8 > 2 means '8 is greater than 2')

≤ : Less than or equal to (e.g., x ≤ 4 means 'x can be 4 or any number less than 4')

≥ : Greater than or equal to (e.g., y ≥ -1 means 'y can be -1 or any number greater than -1')

A simple way to remember the direction is that the symbol is like an alligator's mouth—it always wants to 'eat' the bigger number!

Solving Inequalities

You can solve simple inequalities much like you solve equations. The goal is to isolate the variable. You can add or subtract the same number from both sides without changing the inequality.

Example 1: Solve `x + 5 > 12`

Subtract 5 from both sides:

`x + 5 - 5 > 12 - 5`

`x > 7`

This means the inequality is true for any number greater than 7.

The Big, Important Rule

There is one very important rule you must remember when solving inequalities:

If you multiply or divide both sides of an inequality by a NEGATIVE number, you must FLIP the inequality sign.

Example 2: Solve `-2x < 10`

To isolate x, we need to divide both sides by -2.

Because we are dividing by a negative number, we must flip the `<` sign to a `>` sign.

`(-2x) / (-2) > 10 / (-2)`

`x > -5`

Graphing Inequalities

We can show all the possible solutions to an inequality by graphing it on a number line.

For < and > symbols, we use an open circle on the number line to show that the number itself is not included in the solution.

For ≤ and ≥ symbols, we use a closed circle to show that the number is included.

Then, you shade the part of the number line that represents the possible solutions.

Example: Graph `x ≤ 3`

1.Go to the number 3 on the number line.

2.Draw a closed circle because the symbol is 'less than or equal to'.

3.Shade the number line to the left of the circle, because we want all the numbers that are less than 3.

Key Terms

**Inequality: A mathematical statement that compares two quantities that are not equal, using symbols such as <, >, ≤, or ≥.
**Solution of an Inequality: The set of all numbers that make the inequality true.
**Number Line: A line on which numbers are marked at intervals, used to illustrate simple numerical operations.
**Open Circle: A point on a number line graph that indicates the endpoint is not included in the solution set. Used for < and >.
**Closed Circle: A point on a number line graph that indicates the endpoint is included in the solution set. Used for ≤ and ≥.