Lesson 78: Graphing Inequalities

New Concept

Inequalities use symbols to compare values. The symbol

What’s next

Next, you'll learn to visually represent these relationships on a number line, focusing on the specific use of solid dots versus open circles.

Property

Expressions such as $x \le 4$ and $x > 4$ are called inequalities. They use the symbols $>$, $<$, $\ge$, and $\le$ to describe a range of values.

Examples

Graph of $x > -1$: An empty circle on -1 with a ray pointing to the right.
Graph of $x \le 3$: A solid dot on 3 with a ray pointing to the left.

Explanation

Think of inequalities as describing a whole team of numbers, not just a single player! Instead of saying x equals exactly one thing, you can say 'x is anyone on the team shorter than 4 feet.' Graphing them on a number line gives you a picture of every single number that makes the statement true, which is super useful.

Property

To graph an inequality with $\ge$ (greater than or equal to) or $\le$ (less than or equal to), use a solid dot on the number line to show the number is included.

Examples

Graph of $x \ge 2$: A solid dot on the number 2 with a ray shaded to the right.
Graph of $x \le -1$: A solid dot on the number -1 with a ray shaded to the left.

Explanation

A solid dot is like a VIP ticket—it means that starting number is officially part of the solution set! For $x \ge 4$, the number 4 gets a dot because it is 'equal to 4.' The ray then shows all the other numbers that are 'greater than 4.' It's the whole package: the number itself and everything beyond it.

Property

To graph an inequality with $>$ (greater than) or $<$ (less than), use an empty circle to show the boundary number is not included in the solution.

Examples

Graph of $x > 5$: An empty circle on the number 5 with a ray shaded to the right.
Graph of $x < 0$: An empty circle on the number 0 with a ray shaded to the left.

Explanation

An empty circle is like a fence that you can get right up against but can't touch. For $x > 4$, you can include 4.001 or $4\frac{1}{2}$, but not 4 itself. The empty circle sits right on 4 to say, 'The solutions start right after this spot!' It's the perfect way to exclude a single point.