11-4 Inequalities

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Warm UpSolve.

1. –21z + 12 = –27z

2. –12n – 18 = –6n

3. 12y – 56 = 8y

4. –36k + 9 = –18k

z = –2n = –3

y = 14

Course 2

11-4 Inequalities

k = 12

Problem of the Day

The dimensions of one rectangle are twice as large as the dimensions of another rectangle. The difference in area is 42 cm2. What is the area of each rectangle?

56 cm2 and 14 cm2

Course 2

11-4 Inequalities

Learn to read and write inequalities and graph them on a number line.

Course 2

11-4 Inequalities

Vocabulary

inequalityalgebraic inequalitysolution setcompound inequality

Insert Lesson Title Here

Course 2

11-4 Inequalities

An inequality states that two quantitieseither are not equal or may not be equal. An inequality uses one of the following symbols:

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

Course 2

11-4 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.

Course 2

11-4 Inequalities

Write an inequality for each situation.

Try This: 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.

Course 2

11-4 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.

Course 2

11-4 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

Course 2

11-4 Inequalities

Graph each inequality.

Additional Example 2A & 2B: Graphing Simple Inequalities

–3 –2 –1 0 1 2 3

A. n < 33 is not a solution, so draw an open circle at 3. Shade the line to the left of 3.

B. a ≥ –4

–6 –4 –2 0 2 4 6

–4 is a solution, so draw a closed circleat –4. Shade the lineto the right of –4.

Course 2

11-4 Inequalities

Graph each inequality.

Try This: Example 2A & 2B

–3 –2 –1 0 1 2 3

A. p ≤ 22 is a solution, so draw a closed circle at 2. Shade the line to the left of 2.

B. e > –2

–3 –2 –1 0 1 2 3

–2 is not a solution, so draw an open circleat –2. Shade the lineto the right of –2.

Course 2

11-4 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

Course 2

11-4 Inequalities

Graph each compound inequality.

Additional Example 3A: Graphing Compound Inequalities

0 1 2 3 4 5 6123456– – – – – –

A. m ≤ –2 or m > 1First graph each inequality separately.

m ≤ –2 m > 1

Then combine the graphs.

0 2 4 6246– – –•

0 2 4 6–2–4–6º

The solutions of m ≤ –2 or m > 1 are the combined solutions of m ≤ –2 or m > 1.

Course 2

11-4 Inequalities

Graph each compound inequality

Additional Example 3B: Graphing Compound Inequalities

B. –3 < b ≤ 0–3 < b ≤ 0 can be written as the inequalities–3 < b and b ≤ 0. Graph each inequality separately.

–3 < b b ≤ 0

0 2 4 6246– – –0 2 4 6–2–4–6º •

Then combine the graphs. Remember that –3 < b ≤ 0 means that b is between –3 and 0, and includes 0.

0 1 2 3 4 5 6123456– – – – – –Course 2

11-4 Inequalities

Graph each compound inequality.

Try This: Example 3A

0 1 2 3 4 5 6123456– – – – – –

A. w < 2 or w ≥ 4First graph each inequality separately.

w < 2 W ≥ 4

Then combine the graphs.

0 2 4 6246– – – 0 2 4 6–2–4–6

The solutions of w < 2 or w ≥ 4 are the combined solutions of w < 2 or w ≥ 4.

Course 2

11-4 Inequalities

Graph each compound inequality

Try This: Example 3B

B. 5 > g ≥ –35 > g ≥ –3 can be written as the inequalities5 > g and g ≥ –3. Graph each inequality separately.

5 > g g ≥ –3

0 2 4 6246– – –0 2 4 6–2–4–6º •

Then combine the graphs. Remember that 5 > g ≥ –3 means that g is between 5 and –3, and includes –3.

0 1 2 3 4 5 6123456– – – – – –Course 2

11-4 Inequalities

Lesson Quiz: Part 1

Write an inequality for each situation.

1. No more than 220 people are in the theater.

2. There are at least a dozen eggs left.

3. Fewer than 14 people attended the meeting.

number of eggs ≥ 12

people in the theater ≤ 220

Insert Lesson Title Here

people attending the meeting < 14

Course 2

11-4 Inequalities

Lesson Quiz: Part 2

Graph the inequalities.

4. x > –1

Insert Lesson Title Here

0º

1 3 5123 – – –

5. x ≥ 4 or x < –1

0º

1 3 5135 – – –•

Course 2

11-4 Inequalities