of 4 /4
132 Chapter 3 Solving Linear Inequalities Lesson Tutorials Write each word sentence as an inequality. Graph the inequality. a. A number x is greater than 8 and less than or equal to 4. A number x is greater than 8 and less than or equal to 4. x > 8 and x 4 2 4 6 8 10 12 0 2 4 6 8 b. A number y is at most 0 or at least 7. A number y is at most 0 or at least 7. y 0 or y 7 1 2 4 3 2 1 0 5 6 7 8 EXAMPLE Writing and Graphing Compound Inequalities 1 1 In Exercises 1–4, write the word sentence as an inequality. Graph the inequality. 1. A number k is more than 3 and less than 9. 2. A number n is greater than or equal to 6 and no more than 11. 3. A number w is fewer than 10 or no less than 6. 4. A number z is less than or equal to 5 or more than 4. 5. Write an inequality to describe the graph. 6. The world’s longest human life span is 122 years. Write and graph a compound inequality that describes the ages of all humans. A compound inequality is an inequality formed by joining two inequalities with the word “and” or the word “or.” Solutions of a compound inequality with “and” consist of numbers that are solutions of both inequalities. Solutions of a compound inequality with “or” consist of numbers that are solutions of at least one of the inequalities. x 2 0 1 2 3 4 5 y 2 2 3 1 0 1 2 x < 5 0 1 2 3 4 5 y > 1 2 3 1 0 1 2 2 x and x < 5 y 2 or y > 1 2 3 1 0 1 2 2 x < 5 0 1 2 3 4 5 Key Vocabulary compound inequality, p. 132 absolute value inequality, p. 134 Study Tip A compound inequality with “and” can be written as a single inequality. For example, you can write x > 8 and x 4 as 8 < x 4. 6 5 4 3 2 1 0 1 Solving Compound Inequalities Extension 3.4

# Extension Solving Compound Inequalities 3 · the inequalities − 3 < − 2x + 1 and − 2x + 1 ≤ 9 separately. You can solve compound inequalities by solving two inequalities separately

others

• View
21

1

Embed Size (px)

### Text of Extension Solving Compound Inequalities 3 · the inequalities − 3 < − 2x + 1 and − 2x + 1... 132 Chapter 3 Solving Linear Inequalities

Lesson Tutorials

Write each word sentence as an inequality. Graph the inequality.

a. A number x is greater than −8 and less than or equal to 4.

A number x is greater than − 8 and less than or equal to 4.

x > − 8 and x ≤ 4

24681012 0 2 4 6 8

b. A number y is at most 0 or at least 7.

A number y is at most 0 or at least 7.

y ≤ 0 or y ≥ 7

12 43210 5 6 7 8

EXAMPLE Writing and Graphing Compound Inequalities11

In Exercises 1–4, write the word sentence as an inequality. Graph the inequality.

1. A number k is more than 3 and less than 9.

2. A number n is greater than or equal to 6 and no more than 11.

3. A number w is fewer than − 10 or no less than − 6.

4. A number z is less than or equal to − 5 or more than 4.

5. Write an inequality to describe the graph.

6. The world’s longest human life span is 122 years. Write and graph a compound inequality that describes the ages of all humans.

A compound inequality is an inequality formed by joining two inequalities with the word “and” or the word “or.”

Solutions of a compound inequality with “and” consist of numbers that are solutions of both inequalities.

Solutions of a compound inequality with “or” consist of numbers that are solutions of at least one of the inequalities.

x ≥ 2 0 1 2 3 4 5

y ≤ − 2 23 1 0 1 2

x < 5 0 1 2 3 4 5

y > 1 23 1 0 1 2

2 ≤ x and x < 5 y ≤ − 2 or y > 1 23 1 0 1 22 ≤ x < 5 0 1 2 3 4 5

Key Vocabularycompound inequality, p. 132absolute value inequality, p. 134

Study TipA compound inequality with “and” can be written as a single inequality. For example, you can write x > − 8 and x ≤ 4 as− 8 < x ≤ 4.

6 5 4 3 2 1 0 1

Solving Compound InequalitiesExtension3.4 Extension 3.4 Solving Compound Inequalities 133

Solve −3 < −2x + 1 ≤ 9. Graph the solution.

− 3 < − 2x + 1 ≤ 9 Write the inequality.

− 1 − 1 − 1 Subtract 1 from each expression.

− 4 < − 2x ≤ 8 Simplify.

− 4

— − 2

> − 2x

— − 2

≥ 8 —

− 2 Divide each expression by − 2.

Reverse the inequality symbols.

2 > x ≥ − 4 Simplify.

The solution is − 4 ≤ x < 2.

123456 0 1 2 3 4

EXAMPLE Solving a Compound Inequality with “And”22

Solve the inequality. Graph the solution.

7. 4 < x − 5 < 7 8. − 1 ≤ 2x + 3 < 7 9. 15 > − 3x + 9 ≥ 0

10. 4x + 1 ≤ − 11 or 3x − 4 ≥ 5 11. − 2x − 7 < 5 or − 5x + 6 ≥ 41

Solve 3x − 5 < −8 or 2x − 1 > 5. Graph the solution.

3x − 5 < − 8 or 2x − 1 > 5 Write the inequality.

+ 5 + 5 + 1 + 1 Addition Property of Inequality

3x < − 3 or 2x > 6 Simplify.

3x

— 3

< − 3

— 3

or 2x

— 2

> 6

— 2

Division Property of Inequality

x < − 1 or x > 3 Simplify.

The solution is x < − 1 or x > 3.

1234 43210 5 6

EXAMPLE Solving a Compound Inequality with “Or”33

Study TipYou can also solve the inequality in Example 2 by solving the inequalities

− 3 < − 2x + 1 and

− 2x + 1 ≤ 9 separately.

You can solve compound inequalities by solving two inequalities separately. When a compound inequality with “and” is written as a single inequality, you can solve the inequality by performing the same operation on each expression.

COMMON CORE

Solving Inequalities In this extension, you will● write, solve, and graph

compound inequalities.● write, solve, and graph

absolute value inequalities.Applying StandardsA.CED.1A.CED.3A.REI.3 134 Chapter 3 Solving Linear Inequalities

An absolute value inequality is an inequality that contains an absolute value expression. For example, ∣ x ∣ < 2 and ∣ x ∣ > 2 are absolute value inequalities.

The distance betweenx and 0 is less than 2.

∣ x ∣ < 2

The graph of ∣ x ∣ < 2 is x > − 2 and x < 2.

The distance between x and 0 is greater than 2.

∣ x ∣ > 2

The graph of ∣ x ∣ > 2 is x < − 2 or x > 2.

You can solve these types of inequalities by solving a compound inequality.

Solving Absolute Value Inequalities

To solve ∣ ax + b ∣ < c for c > 0, solve the compound inequality

ax + b > − c and ax + b < c.

To solve ∣ ax + b ∣ > c for c > 0, solve the compound inequality

ax + b < − c or ax + b > c.

In the inequalities above, you can replace < with ≤ and > with ≥ .

a. Solve ∣ x + 7 ∣ ≤ 2. Graph the solution. 2. Graph the solution.

Use ∣ x + 7 ∣ ≤ 2 to write a compound inequality. Then solve.

x + 7 ≥ − 2 and x + 7 ≤ 2 Write compound inequality.

− 7 − 7 − 7 − 7 Subtract 7 from each side.

x ≥ − 9 and x ≤ − 5 Simplify.

The solution is x ≥ − 9 and x ≤ − 5.

b. Solve ∣ 8x − 11 ∣ < 0.

The absolute value of an expression must be greater than or equal to 0. The expression ∣ 8x − 11 ∣ cannot be less than 0.

So, the inequality has no solution.

EXAMPLE Solving Absolute Value Inequalities44

3 2 1 0 1 2 3 3 2 1 0 1 2 3

10 9 8 7 6 5 4

Study TipWhen an absolute value expression is on the left side of an inequality, use an “and” statement for < and ≤, and an “or”statement for > and ≥. Extension 3.4 Solving Compound Inequalities 135

Solve 4 ∣ 2x − 5 ∣ + 1 > 29.

4 ∣ 2x − 5 ∣ + 1 > 29 Write the inequality.

∣ 2x − 5 ∣ > 7 Isolate the absolute value expression.

Use ∣ 2x − 5 ∣ > 7 to write a compond inequality. Then solve.

2x − 5 < − 7 or 2x − 5 > 7 Write compound inequality.

+ 5 + 5 + 5 + 5 Add 5 to each side.

2x < − 2 or 2x > 12 Simplify.

2x

— 2

< − 2

— 2

or 2x

— 2

> 12

— 2

Divide each side by 2.

x < − 1 or x > 6 Simplify.

EXAMPLE Solving an Absolute Value Inequality55

In a poll, 47% of voters say they plan to reelect the mayor. The poll has a margin of error of ±2 percentage points. Write and solve an absolute value inequality to fi nd the least and greatest percents of voters who plan to reelect the mayor.

Words Actual percent of voters

minus percent of voters in poll

is less than or equal to

the margin of error.

Variable Let x represent the actual percent of voters who plan on reelecting the mayor.

Inequality x − 47 ≤ 2

x − 47 ≥ − 2 and x − 47 ≤ 2 Write compound inequality.

+ 47 + 47 + 47 + 47 Add 47 to each side.

x ≥ 45 and x ≤ 49 Simplify.

The least percent of voters who plan to reelect the mayor is 45%. The greatest percent of voters who plan to reelect the mayor is 49%.

EXAMPLE Real-Life Application66

Solve the inequality. Graph the solution, if possible.

12. ∣ x − 3 ∣ ≥ 4 13. ∣ x + 7 ∣ < 1 14. 11 ≥ ∣ 4x − 5 ∣ 15. ∣ 8x − 9 ∣ < 0 16. 3 ∣ 2x + 5 ∣ − 8 ≥ 19 17. − 2 ∣ x − 10 ∣ + 1 > − 7

18. NUMBER SENSE What is the solution of ∣ 4x − 2 ∣ ≥ − 6? Explain.

19. MODELING In Example 6, 44% of the voters say they plan to reelect the mayor. The poll has a margin of error of ±3 percentage points. Use a model to write and solve an absolute value inequality to fi nd the least and greatest percents of voters who plan to reelect the mayor. ##### Inequalities & Regions OCR Module 8. An inequality? An INEQUALITY shows a relationship between two variables, usually x & y Examples –y > 2x + 1 –y
Documents ##### 72006 - Lego72006. 2 3 4. 3 2x 3 25. 4 2x 1x 1x 26 1x 27 1x 1x 28. 5 2x 29 7 2x 30 2x 7 1:1 7 7. 6 31. 7 2x 2x 32 2x. 8 4x 2x 33 1 2 2x. 9 2x 4x 34. 10 2x 2x 35 2x. 11 2x 2x 36. 12
Documents ##### 2x 12x 6060734 3x 368001 2x 6102776 6x 6175968 2x 6123812 4x 6206148 1x 6135130 1x 4143005 2x 6116797 2x 614126 1x 302228 2x 6182261 2x 6094069 4x 6035291 2x 4565394 2x 6146301 1x
Documents ##### CHAPTER 1 Equations, Inequalities, and Mathematical Modeling · 4 x 1 2x 4x 4 2x 2x 4 2 x 2 17. Thus, is an identity by simplification. It is true for all real values of x. x2 8x
Documents ##### Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Aim: How do we solve quadratic inequalities? Do Now: What are the roots for y = x 2 - 2x - 3?
Documents ##### 1.6 Absolute Value Equations and Inequalities. Solving an Absolute Value Equation What is the solution of | 2x – 1 | = 5? Graph the solution. | 2x – 1
Documents ##### Aim: How do we solve Compound Inequalities? Do Now: Solve the following inequalities 1. 2x + 3 > 2 2. 5x < 10 How do we put two inequalities together?
Documents ##### 2015/09/09  · 3y-2x=11 3.3 — Solving Systems of Inequalities Élim. - 6y 12 No Soiofico -2 23. Solve the following systems of inequalities by grap *Check ùesmos* 3.5 & 3.6 —
Documents ##### Pre-Algebra 10-4 Solving Multistep Inequalities. Solve. 1. 6x + 36 = 2x 2. 4x – 13 = 15 + 5x 3. 5(x – 3) = 2x + 3 4. + x = x = –9 x = –28 x = 6 7 8 3
Documents ##### 1.5 Solving Inequalities. Solving and Graphing an Inequality What is the solution of -3(2x – 5) + 1 ≥ 4? Graph the solution. -3(2x – 5) + 1 ≥ 4 -6x
Documents ##### Assembly instructions - Vitavia · 2096 3 619 1x 1x 1x 1x 2097 3 619 1x 1x 1x 1x 2100 2 3 425 2x 2x ... 3041 2 3 1802 2x 2x 2x 2x 3042 2 3 1802 2x 2x 2x 2x 3049 3 1427 2x 2x 2x 2x
Documents ##### 60214...5 17 1x 1x 6 18 2x 7 19 1x 8 20 2x 2x 9 2x 21 2x 10 22 1x 11 23 1x 12 24 2x 13 25 2x 14 26 2x 15 27 2x 16 2x 28 2x 17 29 2x 2x 18 2x 30 2x 19 31 2x 2x 20 2x 32 1x 21 33 1x
Documents ##### 4432 BI.indd 1 10/01/2013 6:04 PM · 4x 300501 2x 300401 2x 300301 2x 301001 2x 407001 2x 4563684 2x 4117061 1x 4620761 2x 4249112 1x 302201 2x 4597131 2x 366601 2x 4598527 4x 4249563
Documents ##### 1 LEGO.com/creator...36 4 37 2x 1x 1 2x 2x 2 2x 1x 3 38 2x 2x 5 2x 2x 2x 6 1x 2x 2x 4 2x 39 26 40 27 41 1x 1x 1 1x 1x 2x 2x 2 1x 42 1x 2x 2x 4 2x 2x 2x 2x 3 43 2x 2x 2x 1x 2x 5 4x
Documents ##### LESSON 5.3 – SYSTEMS OF INEQUALITIES · 212 TOPIC 5 SOLVING LINEAR SYSTEMS. EXPLAIN SOLVING LINEAR SYSTEMS Summary Solving Systems of Linear Inequalities ... y < –2x + 3 4. Graph
Documents ##### 76077 2x 307001 2x 6045952 4x 6020193 371001 2x 6021315 2x 4565452 2x 6013937 2x1x 4666579 2x 6024495 1x 6013081 4x 307021 2x 300521 1x 4142865 2x 6053011 5x 302321 2x 366621 2x 4113858
Documents ##### Chapter 2 Linear Equations and Inequalities in One Variable · The solution set is 12. 6. 3x 6 2x 5 3x 2x 6 2x 2x 5 x 6 5 x 6 6 5 6 x 11 Check: x6 2 5 ( 1)6 2 5 33 6 22 5 27 27 The
Documents ##### Name Date Period - speckmath.net · Solve the following absolute value inequalities. 8. 2x−3 ≥5 9. 2x+4 ≤8 10. x+1 =3 Write the equation of the line with the following conditions
Documents ##### Solve 7x - 3 ≥ 2x + 2 - Mathsbox th.pdf5x + 3 ≤ 2x + 2 Solve x < 7 Solving inequalities 11 6x + 2 ≥ 3x + 3 Solve x > -1 Solving inequalities 12 10x + 2 < 6x - 2 Solve
Documents ##### 70903 - Lego · 78 2x 4504369 2x 6054551 1x 6177269 10x 302301 2x 242001 2x 4249112 6x 6045936 2x 346001 1x 303501 2x 4546705 1x 6175662 2x 6170767 2x 6170768 2x 302328 2x 6170765
Documents ##### High Cut = –6dB High Cut = 0dB - JBL · High Cut = 0dB 1 High Cut = –6dB 2 2x 2x 2x 2x 2x 2x GTO608C 2x 2x 2x 2x 12x 4x 12x 8x 10x 2x 8x 4x 4x 2x +-1 2 3 4 5 2 3 1 1 2 1 2 1 2
Documents ##### 76113...12 20 1x 2x 13 21 4x 14 22 1x 2x 15 1 2 23 1x 1x 16 1 2 3 2 24 4x 2x 2x 17 1 2 3 2x 25 26 2x 18 2x 27 2x 2x 19 1 2 2x 28 2x 20 29 4x 21 30 2x 22 31 2x 23 2x 2x 32 …
Documents ##### 60160...1 2 4 27 2x 22 28 2x 2x 23 29 2x 24 30 1x 25 31 1x 1x 26 32 1x 1x 27 33 1x 2x 28 34 2x 2x 29 35 2x 30 36 2x 31 37 1x 2x 32 38 2x 33 39 2x 2x 34 40 2x 35 41 2x 36 42 4x 2x 37
Documents ##### · 32x 16x 40x 20x 20x 16x 2x 36x 6x 24x 16x 12x 12x 24x 90x 112x 36x 44x 2x 2x 2x 2x 2x 2x 24x 124x 20x 1 Ox 62x 2x 8x 2x 2x 68x 24x 16x 64x 64x
Documents ##### 60080...34 2x 300401 2x 300101 2x 4649167 8x 4558956 2x 6108662 2x 1x 4233486 2x 4143137 1x 389401 1x 4261941 2x 303701 2x 609101 2x 6034044 2x 4581717 4x 241201 2x 371001 1x
Documents ##### 60203...5 1x 4 6 2x 2x 5 1x 6 2x 7 2x 2x 1x 7 2x 8 2x 8 2x 9 2x 9 10 1x 1x 10 2x 11 11 2x 12 12 3x 13 13 2x 14 2x 14 2x 15 1x 1x 16 15 1x 2x 17 16 2x 18 2x 1x 19 17 2x 2x 20 18 4 …
Documents ##### 2x 1x2x 2x 2x 1x 1x 1x - nerd.dschlumpp.comnerd.dschlumpp.com/research_lab_transporter... · 1x 1x1x1x 13 4. 2x 1x 1x 14 2x 2x 2x 2x 15 5. 2x 1x 16 1x 1x 17 6. 1x 2x 3x 18 1x 2x4x
Documents ##### 4432 BI.indd 1 16/01/2013 5:53 PM - Lego · 4x 300501 2x 300401 2x 300301 2x 301001 2x 407001 2x 4563684 2x 4117061 1x 4620761 2x 4249112 1x 302201 2x 4597131 2x 366601 2x 4598527
Documents Documents