Writing and Graphing Inequalities

Because “I<3U” does not mean “I love you.” Mathematically, it means I less than 3 of you.

What is an inequality?

• Mathematical sentence that compares quantities.

• To solve an inequality, you have to find all values for the variable that make the sentence true.

• Inequalities do not contain equal signs.

Some Common Phrases in Inequality Statements

Symbols < > ≤ ≥ ≠Words • less

than• fewer

than• smaller

than• below

• greater than

• more than• Exceeds• larger than• above

• less than or equal to

• no more than

• at most• Maximum• not greater

than

• greater than or equal to

• no less than• at least• Minimum• Not smaller

than

• does not equal

Translating sentences to inequalities

• Example 1: A number is less than 2.

Since we don’t know what the number is, we will use x to represent the unknown number.

x < 2

• Example 2: The cost of the shirt exceeds $15.

x > 15 Exceeds means “is greater than.”

• Example 3: The maximum number of students in the class is 28.

• Example 4: The weight of the package was no less than 50 pounds.

x ≤ 28

x ≥ 50

Maximum means that there could be 28 students or less in the class.

If something is no less than 50, it could be greater than or equal to 50.

To graph an inequality on a number line:

– Draw a number line, and mark the critical point near the center of the number line.

– Frame the critical point by filling in a number to the left and the right.

– Put a circle on the critical point.

• Shade in the circle for inequality symbols ≥ and ≤ .• Leave the circle open (unshaded) for inequality

symbols >, <, and ≠ .– Draw a shaded arrow towards the part of the number line

that makes the inequality statement true.

Examples of Graphing Inequalities

Example 1:

x > 2

Example 2:

x < -6

21 3

-6-7 -5

Example 3:

x ≤ 28

Example 4:

x ≥ -4

2827 29

-4-5 -3