8.2 Graphing Inequalities
Learning Objectives
- I can express inequalities in various forms, as well as graph inequalities on a number line.
Introduction
Suppose you're having a party, and you know that the number of people attending will be greater than or equal to 25. How would you write this inequality? If you had to graph the solutions to this inequality on a number line, could you do it? After completing this concept, you'll not only be able to express inequalities such as this one with a graph, but you'll also be able to look at a graph and determine what inequality it represents.
Guided Learning
Verbs that translate into inequalities are:
“greater than”
“greater than or equal to”
“less than”
“less than or equal to”
“not equal to”
An algebraic inequality is a mathematical sentence connecting an expression to a value, a variable, or another expression with an inequality sign.
Solutions to one-variable inequalities can be graphed on a number line or in a coordinate plane.
Example A
Graph the solutions to on a number line.
The inequality is asking for all real numbers larger than three, but not including three. This is shown on the number line by using an open circle on the number three. This means that the graph does not include three as a solution. The next step is to draw a line in the direction of your solution. If the inequality is less than, your line will extend to the left as numbers on the number line become smaller. If the inequality is greater than, your line will extend to the right as numbers on the number line become larger. For this problem, t is greater than three, so the line extends to the right.
You can also write inequalities given a number line of solutions.
Example B
Write the inequality pictured below.
The value of four is colored in, meaning that four is a solution to the inequality. The red arrow indicates values less than four. Therefore, the inequality is:
Inequalities that “include” the value are shown as or . The line underneath the inequality stands for “or equal to.” We show this relationship by coloring in the circle above this value on the number line, as in the previous example. For inequalities without the “or equal to,” the circle above the value on the number line remains unfilled.
Ways to Express Solutions to Inequalities
1. Inequality notation: The answer is expressed as an algebraic inequality, such as .
2. Graph sentences on a number line.
Take a minute to write a note in your journal about graphing inequalities.
Representing inequalities on a number line.
First determine if you should use a closed circle or an open circle on the graph.
A closed circle indicates .
An open circle indicates .
Draw a line from the number used in the direction for less than (to the left, small numbers) or greater than (to the right, larger numbers).
Guided Practice
1. Graph the solution expressed by .
2. Write the inequality shown by the graph.
Check your work with a partner.
Solutions:
1. The solution contains all numbers less than 3.25, not including 3.25. We show this by using an open circle on the graph.
2. The number 1 is included in the solution since an closed circle is used on the graph.
.
Review
- Greater than or equal to & less than or equal to are shown with a closed circle on the number line.
- Greater than & less than are shown with an open circle on the number line.
Algebraic inequality
A mathematical sentence connecting an expression to a value, a variable, or another expression with an inequality sign.