1-10 introduction to inequalities warm up warm up lesson presentation lesson presentation problem of...

1-10 Introduction to Inequalities

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Problem of the DayProblem of the Day

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1-10 Introduction to Inequalities

Warm UpSolve.

1. x + 6 = 132. 8n = 483. t – 2 = 564. 6 =

x = 7

n = 6t = 58

z = 36z6

1-10 Introduction to Inequalities

Problem of the Day

Bill and Brad are taking Drivers Education class. Bill drives with his instructor for one and a half hours three times a week. He needs a total of 27 hours. Brad drives two times a week, two hours each time. He needs 26 hours. Who will finish his hours first? Bill

1-10 Introduction to Inequalities

Learn to solve and graph inequalities.

1-10 Introduction to Inequalities

Vocabularyinequalityalgebraic inequalitysolution set

1-10 Introduction to Inequalities

An inequality compares two quantities and typically uses one of these symbols:

<<is less than

is greater than

is less than or equal to

is greater than or equal to

1-10 Introduction to Inequalities

The inequality symbol opens to the side with the greater number.

2 < 10

Remember!

1-10 Introduction to Inequalities

Additional Example 1: Completing an Inequality

Compare. Write < or >.

A. 23 – 14 6

9 6>

B. 5(12) 70

60 70<

1-10 Introduction to Inequalities

Check It Out: Example 1

Compare. Write < or >.

A. 19 – 3 17

16 17<

B. 4(15) 50

60 50>

1-10 Introduction to Inequalities

An inequality that contains one or more variables is an algebraic inequality.

A number that makes an inequality true is a solution of the inequality.

The set of all solutions is called the solution set. The solution set can be shown by graphing it on a number line.

1-10 Introduction to Inequalities

x < 5

4 < 5x = 2.1 2.1 < 5

x is less than 5Word

Phrase

Inequality

Sample Solutions

Solution Set 1 2 3 4 5 6 7

x = 4

1-10 Introduction to Inequalities

a > 0

7 > 0a = 25 25 > 0

a is greater than 0

a is more than 0Word

Phrase

Inequality

Sample Solutions

Solution Set–3 –2 –1 0 1 2 3

a = 7

1-10 Introduction to Inequalities

y 2

0 2y = 1.5 1.5 2

y is less than or equal to 2

y is at most 2Word

Phrase

Inequality

Sample Solutions

Solution Set–3 –2 –1 0 1 2 3

y = 0

1-10 Introduction to Inequalities

m 3

17 3m = 3 3 3

m is greater than or equal to 3

m is at least 3Word

Phrase

Inequality

Sample Solutions

Solution Set–1 0 1 2 3 4 5

m = 17

1-10 Introduction to Inequalities

Most inequalities can be solved the same way equations are solved.

Use inverse operations on both sides of the inequality to isolate the variable.

1-10 Introduction to Inequalities

An open circle means that the corresponding value is not a solution. A solid circle means that the value is part of the solution set.

Helpful Hint!

1-10 Introduction to Inequalities

Additional Example 2A: Solving and Graphing Inequalities

Solve and graph the inequality.

x + 2.5 8 –2.5 –2.5

x 5.5

1 2 3 4 5 6 7

Use the Subtraction Property of Inequality:Subtract 2.5 from both sides.

According to the graph, 5.4 is a solution, since 5.4 < 5.5, and 6 should not be solution because 6 > 5.5.

1-10 Introduction to Inequalities

Additional Example 2A Continued

Check

Substitute 5.4 for x.

7.9< 8 ?

So 5.4 is a solution.

x + 2.5 < 8 ?

5.4 + 2.5 < 8 ?

Check

Substitute 6 for x.

8.5< 8 ?

So 6 is not a solution.

x + 2.5 < 8 ?

6 + 2.5 < 8 ?

1-10 Introduction to Inequalities

Additional Example 2B: Solving and Graphing Inequalities

Solve and graph the inequality.

w – 1 < 8

w < 9

–3 0 3 6 9 12 15

+ 1 + 1 Use the Addition Property of Inequality: Add 1 to both sides.

1-10 Introduction to Inequalities

Check It Out: Example 2

Solve and graph each inequality.

A. x + 2 3.5 –2 –2x 1.5

1 2 3 4 5 6 7

Use the Subtraction Property of Inequality: Subtract 2 from both sides.

B. 6u > 72

6 6

u > 12 3 6 9 12 15 18 21

6u > 72 Use the Division Property of Inequality: Divide both sides by 6.

1-10 Introduction to Inequalities

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

1-10 Introduction to Inequalities

Lesson Quiz

Compare. Use < or > to compare each inequality.

1. 13 5(2) 2. 14 – 2 11

Solve and graph each inequality.

3. k + 9 < 12

4. m – 4 2

5. A school bus can hold no more than 64 passengers. There are already 21 passengers on the bus. Write and solve an inequality to find how many more passengers the bus can hold.

>

m 6

>

k < 3–5 –4–3–2–1 0 1 2 3 4 5

–4 –3–2–1 0 1 2 3 4 5 6

x + 21 64; x 43

1-10 Introduction to Inequalities

1. Identify the correct sign to compare.

19 4(4)

A. >

B. <

C. ≥

D. ≤

Lesson Quiz for Student Response Systems

1-10 Introduction to Inequalities

2. Identify the correct sign to compare.

19 – 3 21

A. >

B. <

C. ≤

D. ≥

Lesson Quiz for Student Response Systems

1-10 Introduction to Inequalities

3. Solve and graph the inequality.

x + 7 < 10

A. x < 3

B. x < 4

Lesson Quiz for Student Response Systems

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10

1-10 Introduction to Inequalities

4. Solve and graph the inequality.

4 ≤

A. 8 ≤ a

B. 6 ≤ a

Lesson Quiz for Student Response Systems

a2

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10

1-10 Introduction to Inequalities

5. An egg tray can hold 4 dozen eggs. Two friends have 24 eggs each. They would like to put all the eggs in the tray. Write and solve an inequality to determine whether all the eggs will fit in the tray.

A. 2(24) ≤ 4; no

B. 2(24) ≤ 48; yes

C. 2(48) ≤ 24; yes

D. 2(24) ≤ 24; no

Lesson Quiz for Student Response Systems