Holt CA Course 1

11-5 Introduction to Inequalities

Preview of Grade 7 AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations, or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).

California Standards

Holt CA Course 1

11-5 Introduction to Inequalities

Vocabulary

inequalityalgebraic inequalitysolution setcompound inequality

Holt CA Course 1

11-5 Introduction to Inequalities

An inequality is a statement that compares two expressions by using one of the following symbols: <, >, ≤, ≥, or .

Symbol Meaning Word Phrases

<

>

≤

≥

is less than

is greater than

is greater than or equal to

is less than or equal to

Fewer than, below

More than, above

At most, no more than

At least, no less than

Holt CA Course 1

11-5 Introduction to Inequalities

Write an inequality for each situation.

Additional Example 1: Writing Inequalities

A. There are at least 15 people in the waiting room.

number of people ≥ 15

B. The tram attendant will allow no more than 60 people on the tram.

number of people ≤ 60

“At least” means greaterthan or equal to.

“No more than” meansless than or equal to.

Holt CA Course 1

11-5 Introduction to Inequalities

Write an inequality for each situation.

Check It Out! Example 1

A. There are at most 10 gallons of gas in the tank.

gallons of gas ≤ 10

B. There is at least 10 yards of fabric left.

yards of fabric ≥ 10

“At most” means lessthan or equal to.

“At least” meansgreater than or equal to.

Holt CA Course 1

11-5 Introduction to Inequalities

An inequality that contains a variable is an algebraic inequality. A value of the variable that makes the inequality true is a solution of the inequality.

An inequality may have more than one solution. Together, all of the solutions are called the solution set.

You can graph the solutions of an inequality on a number line. If the variable is “greater than” or “less than” a number, then that number is indicated with an open circle.

Holt CA Course 1

11-5 Introduction to Inequalities

This open circle shows that 5 is not a solution.

a > 5

If the variable is “greater than or equal to” or “less than or equal to” a number, that number is indicated with a closed circle.

This closed circle shows that 3 is a solution.

b ≤ 3

Holt CA Course 1

11-5 Introduction to Inequalities

Graph each inequality.

Additional Example 2: Graphing Simple Inequalities

–3 –2 –1 0 1 2 3

A. n < 3 Draw an open circle at 3. The solutions are values of n less than 3, so shade to the left of 3.

B. a ≥ –4

–6 –4 –2 0 2 4 6

Draw a closed circleat –4. The solutions are –4 and values of a greater than –4, so shade to the right of –4.

Holt CA Course 1

11-5 Introduction to Inequalities

Graph each inequality.

Check It Out! Example 2

–3 –2 –1 0 1 2 3

A. p ≤ 2 Draw a closed circle at 2. The solutions are 2 and values of p less than 2, so shade to the left of 2.

B. e > –2

–3 –2 –1 0 1 2 3

Draw an open circleat –2. The solutions are values of e greater than –2, so shade to the right of –2.

Holt CA Course 1

11-5 Introduction to Inequalities

A compound inequality is the result of combining two inequalities. The words and and or are used to describe how the two parts are related.

x > 3 or x < –1 –2 < y and y < 4

x is either greater than 3 or less than–1.

y is both greater than –2 and less than 4. y is between –2 and 4.

The compound inequality –2 < y and y < 4 can be written as –2 < y < 4.

Writing Math

Holt CA Course 1

11-5 Introduction to Inequalities

Graph each compound inequality.

Additional Example 3: Graphing Compound Inequalities

0 1 2 3 4 5 6123456– – – – – –

A. m ≤ –2 or m > 1

Graph m ≤ –2 Graph m > 1

Include the solutions shown by either graph.

0 2 4 6246– – –•

0 2 4 6–2–4–6º

Holt CA Course 1

11-5 Introduction to Inequalities

Graph each compound inequality.

Additional Example 3: Graphing Compound Inequalities

B. –3 < b ≤ 0

Graph –3 < b Graph b ≤ 0

0 2 4 6246– – –0 2 4 6–2–4–6º •

Include the solutions that the graphs have in common.

0 1 2 3 4 5 6123456– – – – – –

Holt CA Course 1

11-5 Introduction to Inequalities

Graph each compound inequality.

Check It Out! Example 3

0 1 2 3 4 5 6123456– – – – – –

A. w < 2 or w ≥ 4

Graph w < 2 Graph w ≥ 4

Include the solutions shown by either graph.

0 2 4 6246– – – 0 2 4 6–2–4–6

Holt CA Course 1

11-5 Introduction to Inequalities

Graph each compound inequality.

Check It Out! Example 3

B. 5 > g ≥ –3

Graph 5 > g Graph g ≥ –3

0 2 4 6246– – –0 2 4 6–2–4–6º •

Include the solution that the graphs have in common.

0 1 2 3 4 5 6123456– – – – – –