182
Slide 1 / 182

Solving and Graphing Linear Inequalities

  • Upload
    others

  • View
    7

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Solving and Graphing Linear Inequalities

Slide 1 / 182

Page 2: Solving and Graphing Linear Inequalities

Slide 2 / 182

Algebra I

Solving & Graphing Inequalities

2016-01-11

www.njctl.org

Page 3: Solving and Graphing Linear Inequalities

Slide 3 / 182

Table of Contents

Simple Inequalities Addition/Subtraction

Simple Inequalities Multiplication/Division

Solving Compound Inequalities

Special Cases of Compound Inequalities

Graphing Linear Inequalities in Slope-Intercept Form

click on the topic to go to that section

Glossary & Standards

Solving Systems of Inequalitites

Two-Step and Multiple-Step Inequalities

Page 4: Solving and Graphing Linear Inequalities

Slide 4 / 182

Simple Inequalities Involving Additionand Subtraction

Return to Table of Contents

Page 5: Solving and Graphing Linear Inequalities

Slide 5 / 182

An Inequality is a mathematical sentence that uses symbols, such as <, ≤, > or ≥ to compare to quantities.

Inequality

Page 6: Solving and Graphing Linear Inequalities

Slide 6 / 182

What do these symbols mean?

LessThan

Less Than or Equal To

Greater Than

Greater Thanor Equal To

click

click

(when read from LEFT to RIGHT)

Page 7: Solving and Graphing Linear Inequalities

Slide 7 / 182

Page 8: Solving and Graphing Linear Inequalities

Slide 8 / 182

Write an inequality for the sentence below:

The sum of a number, n, and fifteen is greater than or equal to nine.

Three times a number, n, is less than 210.

Inequality

Click

Click

Page 9: Solving and Graphing Linear Inequalities

Slide 9 / 182

Remember!

Open circle means that number is not included in the solution set and is used to represent < or >.

Closed circle means the solution set includes that number and is used to represent ≤ or ≥.

Graphing Inequalities

Page 10: Solving and Graphing Linear Inequalities

Slide 10 / 182

· Solving one-step inequalities is much like solving one-step equations.

· To solve an inequality, you need to isolate the variable using the properties of inequalities and inverse operations.

Solving Inequalities

Page 11: Solving and Graphing Linear Inequalities

Slide 11 / 182

To find the solution, isolate the variable x.Remember, it is isolated when it appears by itself on one side of the equation.

Isolate the Variable

Page 12: Solving and Graphing Linear Inequalities

Slide 12 / 182

Page 13: Solving and Graphing Linear Inequalities

Slide 13 / 182

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Step 2: Decide whether or not the circle on your boundary should be open or closed based on the symbol used. You can check the computation by substituting the end point of 6 for x. In this case, the end point is not included (open circle) since x < 6.

Solving Inequalities

Page 14: Solving and Graphing Linear Inequalities

Slide 14 / 182

Page 15: Solving and Graphing Linear Inequalities

Slide 15 / 182

Review of Solving Inequalities Using Addition and Subtraction

The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at:

http://www.njctl.org/courses/math/7th-grade/equations-inequalities-7th-grade/

Page 16: Solving and Graphing Linear Inequalities

Slide 16 / 182

0 1 2 3 4 5-1-2-3-4-5

2 5 6

0 1 2 3 4 5-1-2-3-4-5

2 5 6

0 1 2 3 4 5-1-2-3-4-5

2 5 6

0 1 2 3 4 5-1-2-3-4-5

5 62

A

B

C

D

1 Which graph is the solution to the inequality: a number, n, minus is greater than one third?

Page 17: Solving and Graphing Linear Inequalities

Slide 17 / 182

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

A

B

C

D

2 Which graph is the solution to the inequality ?

Page 18: Solving and Graphing Linear Inequalities

Slide 18 / 182

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10A

B

C

D

3 Which graph is the solution to the inequality ?

Page 19: Solving and Graphing Linear Inequalities

Slide 19 / 182

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10A

B

C

D

4 Which graph is the solution to the inequality ?

Page 20: Solving and Graphing Linear Inequalities

Slide 20 / 182

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1.510 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1.510 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1.510 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1.5A

B

C

D

5 Which graph is the solution to the inequality ?

Page 21: Solving and Graphing Linear Inequalities

Slide 21 / 182

Simple Inequalities Involving Multiplication

and Division

Return to Table of Contents

Page 22: Solving and Graphing Linear Inequalities

Slide 22 / 182

Again, similarly to solving equations, we can use the properties of multiplication and division to solve and graph inequalities - with one minor difference, which we will encounter in the upcoming slides.

Inequalities Involving Multiplication and Division

Page 23: Solving and Graphing Linear Inequalities

Slide 23 / 182

Since x is multiplied by 3, divide both sides by 3 to isolate the variable.

Multiplying or Dividing by a Positive Number

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 24: Solving and Graphing Linear Inequalities

Slide 24 / 182

Page 25: Solving and Graphing Linear Inequalities

Slide 25 / 182

Review of Solving Inequalities Using Multiplication and Division

The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at:

http://www.njctl.org/courses/math/7th-grade/equations-inequalities-7th-grade/

Page 26: Solving and Graphing Linear Inequalities

Slide 26 / 182

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

6 Which graph is the solution to the inequality, the product of 4 and a number, x, is greater than 24?

A

B

C

D

Page 27: Solving and Graphing Linear Inequalities

Slide 27 / 182

Page 28: Solving and Graphing Linear Inequalities

Slide 28 / 182

Page 29: Solving and Graphing Linear Inequalities

Slide 29 / 182

Find the solution to the inequality.9

A

B

C

D

Page 30: Solving and Graphing Linear Inequalities

Slide 30 / 182

10

A

B

C

Find the solution to the inequality.

D

Page 31: Solving and Graphing Linear Inequalities

Slide 31 / 182

So far, all the operations we have used worked the same as solving equations. The difference between solving

equations versus inequalities is revealed when multiplying or dividing by a negative number.

The direction of the inequality changes only if the number you are using to multiply or divide by is

negative.

Multiplying or Dividing by a Negative Number

Page 32: Solving and Graphing Linear Inequalities

Slide 32 / 182

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

*Note: Dividing each side by -3 changes the ≥ to ≤.

Solve and Graph

click for answer

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 33: Solving and Graphing Linear Inequalities

Slide 33 / 182

11 Solve the inequality and graph the solution.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 34: Solving and Graphing Linear Inequalities

Slide 34 / 182

12

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve the inequality and graph the solution.

Page 35: Solving and Graphing Linear Inequalities

Slide 35 / 182

13

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve the inequality and graph the solution.

Page 36: Solving and Graphing Linear Inequalities

Slide 36 / 182

14

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve the inequality and graph the solution.

Page 37: Solving and Graphing Linear Inequalities

Slide 37 / 182

In review, an inequality symbol stays the same direction when you:

· Add, subtract, multiply or divide by the same positive number on both sides.

· Add or subtract the same negative number on both sides.

An inequality symbol changes direction when you:

· Multiply or divide by the same negative number on both sides.

Summary

Page 38: Solving and Graphing Linear Inequalities

Slide 38 / 182

Solving Two-Step and Multiple-Step

InequalitiesReturn to Table of Contents

Page 39: Solving and Graphing Linear Inequalities

Slide 39 / 182

Now we'll solve more complicated inequalities that have multi-step solutions because they involve more than one operation.

Solving inequalities is like solving a puzzle. Keep working through the steps until you get the variable you're looking for alone on one side of the inequality using the same strategies as solving an equation.

Inequalities

Page 40: Solving and Graphing Linear Inequalities

Slide 40 / 182

Page 41: Solving and Graphing Linear Inequalities

Slide 41 / 182

Another reminder! If you multiply or divide by a negative number, reverse the

direction of the inequality symbol!

Multiplying or Dividing by a Negative Number

Page 42: Solving and Graphing Linear Inequalities

Slide 42 / 182

Page 43: Solving and Graphing Linear Inequalities

Slide 43 / 182

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Add 9 to both sides

Divide both sides by 4(sign stays the same)

Example: Solve the inequality and graph the solution.

Two Step Inequalities

click for answer

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 44: Solving and Graphing Linear Inequalities

Slide 44 / 182

Try these.Solve each inequality and graph each solution.

1.

2.

Solve and Graph

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 45: Solving and Graphing Linear Inequalities

Slide 45 / 182

Try these. Solve each inequality and graph the solution.

3.

4.

Solve and Graph

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 46: Solving and Graphing Linear Inequalities

Slide 46 / 182

15

A

B

C

Solve and graph the solution.

D

0 1 2 3 4 5-1-2-3-4-5

2.5

0 1 2 3 4 5-1-2-3-4-5

2.5

0 1 2 3 4 5-1-2-3-4-5

2.5

0 1 2 3 4 5-1-2-3-4-5

2.5

Page 47: Solving and Graphing Linear Inequalities

Slide 47 / 182

Page 48: Solving and Graphing Linear Inequalities

Slide 48 / 182

Page 49: Solving and Graphing Linear Inequalities

Slide 49 / 182

Page 50: Solving and Graphing Linear Inequalities

Slide 50 / 182

19

A

B

C

Solve and graph the solution.

D

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 51: Solving and Graphing Linear Inequalities

Slide 51 / 182

20 Which graph represents the solution set for:

A

B

C

D

Question from ADP Algebra I End-of-Course Practice Test

0 1 2-1-2

0 1 2-1-2

0 1 2-1-2

0 1 2-1-2

Page 52: Solving and Graphing Linear Inequalities

Slide 52 / 182

21

A

B

C

D

E

F

G

H

Find all negative odd integers that satisfy the following inequality. Select all that apply.

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 53: Solving and Graphing Linear Inequalities

Slide 53 / 182

22 Which value of x is in the solution set of ?

A 8B 9C 12D 16

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 54: Solving and Graphing Linear Inequalities

Slide 54 / 182

23 What is the solution of ?

ABCD

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 55: Solving and Graphing Linear Inequalities

Slide 55 / 182

24 In the set of positive integers, what is the solution set of the inequality ?

ABCD

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 56: Solving and Graphing Linear Inequalities

Slide 56 / 182

Page 57: Solving and Graphing Linear Inequalities

Slide 57 / 182

26 Given: Determine all elements of set A that are in the solution of the inequality .

A 18

B 6

C -3D -12

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 58: Solving and Graphing Linear Inequalities

Slide 58 / 182

Inequalities in the Real World

Inequalities are helpful when applied to real life scenarios. These inequalities can be used for budgeting purposes, speed limits, cell phone data usage, and building materials management, just to name a few.

Translating between the languages of English words to numbers/symbols is imperative in being able to solve the correct inequality. The next slides will provide ample practice in setting up and solving these inequality applications.

Page 59: Solving and Graphing Linear Inequalities

Slide 59 / 182

Page 60: Solving and Graphing Linear Inequalities

Slide 60 / 182

Example #2: You have $65.00 in birthday money and want to buy some CDs and a DVD. Suppose a DVD cost $15.00 and a CD cost $12.00.

Write an inequality and solve to find out how many CDs you can buy along with one DVD.

Write an Inequality and Solve

Page 61: Solving and Graphing Linear Inequalities

Slide 61 / 182

Example #3: Matt was getting ready to go back to school. He had $150 to buy school supplies. Matt bought 3 pairs of pants and spent $30 on snacks and other items.

How much could one pair of pants cost, if they were all the same price? Write an inequality and solve.

Write an Inequality and Solve

Page 62: Solving and Graphing Linear Inequalities

Slide 62 / 182

Example #4: You have $60 to spend on a concert. Tickets cost $18 each and parking is $8. Write an inequality to model the situation. How many tickets can you buy?

Write an Inequality and Solve

Page 63: Solving and Graphing Linear Inequalities

Slide 63 / 182

Example #5: If you borrow the $60 from your mom and pay her back at a rate of $7 per week, when will your debt be under $15?

Write an Inequality and Solve

Page 64: Solving and Graphing Linear Inequalities

Slide 64 / 182

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Example #6: To earn an A in math class, you must earn a total of at least 180 points on three tests. On the first two tests, your scores were 58 and 59. What is the minimum score you must get on the third test in order to earn an A?

Define a variable, write an inequality and graph the solutions.

Write an Inequality and Solve

Page 65: Solving and Graphing Linear Inequalities

Slide 65 / 182

Example #7: Thelma and Laura start a lawn-mowing business and buy a lawnmower for $225. They plan to charge $15 to mow one lawn. What is the minimum number of lawns they need to mow if they wish to earn a profit of at least $750?

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

Write an Inequality and Solve

Page 66: Solving and Graphing Linear Inequalities

Slide 66 / 182

27 Roger is having a picnic for 78 guests. He plans to serve each guest at least one hot dog. If each package, p, contains eight hot dogs, which inequality could be used to determine how many packages of hot dogs Roger will need to buy?

ABCD

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 67: Solving and Graphing Linear Inequalities

Slide 67 / 182

28 A school group needs a banner to carry in a parade. The narrowest street the parade is marching down measures 36 ft across, but some space is taken up by parked cars. The students have decided the banner should be 18 ft long. There is 45 ft of trim available to sew around the border of the banner. What is the greatest possible width for the banner?

A

B

C

D

Page 68: Solving and Graphing Linear Inequalities

Slide 68 / 182

29 Admission to a town fair is $7.00. You plan to spend $6.00 for lunch and $4.50 for snacks. Each ride costs $2.25. If you have $35 to spend, what is the number of rides you can go on?

A

B

C

D

6 rides

7 rides

8 rides

9 rides

Page 69: Solving and Graphing Linear Inequalities

Slide 69 / 182

30 A female gymnast is participating in a 4-event competition. Each event is scored on a ten-point scale. She scored a 9.1 in uneven bars, an 8.5 on the balance beam, and a 9.4 on the vault. Which inequality represents the remaining score required in the floor exercise for the gymnast to receive at least an 8.9 average?

A r ≥ 8.975

B r ≥ 8.6

C r ≤ 8.975

D r ≤ 8.6

Page 70: Solving and Graphing Linear Inequalities

Slide 70 / 182

Solving CompoundInequalities

Return to Table of Contents

Page 71: Solving and Graphing Linear Inequalities

Slide 71 / 182

Compound Inequalities

When two inequalities are combined into one statement by the words AND/OR, the result is called a compound inequality.

A solution of a compound inequality joined by and is any number that makes both inequalities true.

A solution of a compound inequality joined by or is any number that makes either inequality true.

Page 72: Solving and Graphing Linear Inequalities

Slide 72 / 182

Page 73: Solving and Graphing Linear Inequalities

Slide 73 / 182

Page 74: Solving and Graphing Linear Inequalities

Slide 74 / 182

0 1 2 3 4 5-1-2-3-4-5

31 Which inequality is represented in the graph below?

AB

C

D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 75: Solving and Graphing Linear Inequalities

Slide 75 / 182

0 1 2 3 4 5-1-2-3-4-5

32 Which inequality is represented in the graph below?

A

B

C

D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 76: Solving and Graphing Linear Inequalities

Slide 76 / 182

is the same as writing

AND

You will need to solve both of these inequalities and graph their intersection.

Solving Compound Inequalities that contain an AND statement

Page 77: Solving and Graphing Linear Inequalities

Slide 77 / 182

Page 78: Solving and Graphing Linear Inequalities

Slide 78 / 182

Page 79: Solving and Graphing Linear Inequalities

Slide 79 / 182

33 Which result below is correct for this inequality:

A

B

C

0 1 2 3 4 5-1-2-3-4-5

0 1 2 3 4 5-1-2-3-4-5

0 1 2 3 4 5-1-2-3-4-5

Page 80: Solving and Graphing Linear Inequalities

Slide 80 / 182

34 Which result below is correct for this inequality:

A

B

C

0 1 2 3 4 5-1-2-3-4-5

2 1/2

0 1 2 3 4 5-1-2-3-4-5

2 1/2

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 81: Solving and Graphing Linear Inequalities

Slide 81 / 182

35 Which result below is correct for this inequality:

A

B

C

Page 82: Solving and Graphing Linear Inequalities

Slide 82 / 182

36 Which result below is correct for this inequality:

A

B

C

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 83: Solving and Graphing Linear Inequalities

Slide 83 / 182

37 Which result below is correct for this inequality:

A

B

C

Page 84: Solving and Graphing Linear Inequalities

Slide 84 / 182

Page 85: Solving and Graphing Linear Inequalities

Slide 85 / 182

Page 86: Solving and Graphing Linear Inequalities

Slide 86 / 182

Writing a Compound Inequality From a Graph

How would you write this?

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 87: Solving and Graphing Linear Inequalities

Slide 87 / 182

Writing a Compound Inequality From a Graph

How would you write this?

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 88: Solving and Graphing Linear Inequalities

Slide 88 / 182

1.

2. or

Compound InequalitiesSolve and graph the solution set.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 89: Solving and Graphing Linear Inequalities

Slide 89 / 182

3. or

4.

Compound InequalitiesSolve and graph the solution set.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Page 90: Solving and Graphing Linear Inequalities

Slide 90 / 182

38 In order to be admitted for a certain ride at an amusement park, a child must be greater than or equal to 36 inches tall and less than 48 inches tall. Which graph represents these conditions?

A

B

C

D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

3635 37 38 39 40 41 42 43 44 45 46 47 48 49 50

3635 37 38 39 40 41 42 43 44 45 46 47 48 49 50

3635 37 38 39 40 41 42 43 44 45 46 47 48 49 50

3635 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Page 91: Solving and Graphing Linear Inequalities

Slide 91 / 182

Page 92: Solving and Graphing Linear Inequalities

Slide 92 / 182

40 Which graph represents the solution set for #### and ?

10 2 3 4 5 6 7 8 9 10 111213 1415 16171819 20A

10 2 3 4 5 6 7 8 9 10 111213 1415 16171819 20B

10 2 3 4 5 6 7 8 9 10 111213 1415 16171819 20C

10 2 3 4 5 6 7 8 9 10 111213 1415 16171819 20D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 93: Solving and Graphing Linear Inequalities

Slide 93 / 182

41 Solve

A

B

C

D

Page 94: Solving and Graphing Linear Inequalities

Slide 94 / 182

Page 95: Solving and Graphing Linear Inequalities

Slide 95 / 182

Page 96: Solving and Graphing Linear Inequalities

Slide 96 / 182

Page 97: Solving and Graphing Linear Inequalities

Slide 97 / 182

Page 98: Solving and Graphing Linear Inequalities

Slide 98 / 182

Application of Compound Inequalities

Let's start off by translating the words of an applied problem into math.

The sum of 3 times a number and two lies between 8 and 11.

"The sum of 3 times a number and two" translates into what?

Page 99: Solving and Graphing Linear Inequalities

Slide 99 / 182

The sum of 3 times a number and two lies between 8 and 11.

How will we translate "lies between 8 and 11"?

What inequality symbol will we use? Why?

What is the inequality? Solve and graph the inequality.

Application of Compound Inequalities

Page 100: Solving and Graphing Linear Inequalities

Slide 100 / 182

A cell phone plan offers free minutes for no more than 250 minutes per month. Define a variable and write an inequality for the possible number of free minutes. Graph the solution.

Application of Compound Inequalities

Page 101: Solving and Graphing Linear Inequalities

Slide 101 / 182

46 Each type of marine mammal thrives in a specific range of temperatures. The optimal temperatures for dolphins range from 50°F to 90°F. Which inequality represents the temperatures where dolphins will not thrive?

A

B

C

D

Page 102: Solving and Graphing Linear Inequalities

Slide 102 / 182

Page 103: Solving and Graphing Linear Inequalities

Slide 103 / 182

48 A store is offering a $50 mail in rebate on all color printers. Nathan is looking at different color printers that range in price from $165 to $275. How much can he expect to spend after the rebate?

A $115 ≤ x ≤ $225

B x < $115 or x > $225

C $215 ≤ x ≤ $325

D x < $215 or x > $325

Page 104: Solving and Graphing Linear Inequalities

Slide 104 / 182

49 One quarter of a number decreased by 7 is at most 11 or greater than 15. Which compound inequality represents the possible values of the number?

A

B

C

D

Page 105: Solving and Graphing Linear Inequalities

Slide 105 / 182

50 Lyla has scores of 82, 92, 93, and 99 on her math tests. Use a compound inequality to find the range of scores she can make on her final exam to receive a B in the course. The final exam counts as two test grades, and a B is received if the final course average is from 85 to 92.

A

B

C

D

Page 106: Solving and Graphing Linear Inequalities

Slide 106 / 182

Special Cases of Compound Inequalities

Return to Table of Contents

Page 107: Solving and Graphing Linear Inequalities

Slide 107 / 182

Special Cases

A solution of a compound inequality joined by and is any number that makes both inequalities true.

When there is no number that makes both inequalities true, we say there is no solution.

When all numbers make both inequalities true, we say the solution is the set of Reals or All Reals.

Page 108: Solving and Graphing Linear Inequalities

Slide 108 / 182

Page 109: Solving and Graphing Linear Inequalities

Slide 109 / 182

Page 110: Solving and Graphing Linear Inequalities

Slide 110 / 182

Solve each set of compound inequalities.

1. and

2. or

Special Cases

Page 111: Solving and Graphing Linear Inequalities

Slide 111 / 182

Solve each set of compound inequalities.

3. and

4. and

Special Cases

Page 112: Solving and Graphing Linear Inequalities

Slide 112 / 182

Graphing Linear Inequalitiesin Slope-Intercept Form

Return to Table of Contents

Page 113: Solving and Graphing Linear Inequalities

Slide 113 / 182

Page 114: Solving and Graphing Linear Inequalities

Slide 114 / 182

The following are graphs of linear inequalities.

Shading is above the dotted line.This means the solutions are above the line but NOT on it.

Shading is below the dotted line.This means the solutions are below the line but NOT on it.

Graphing

Page 115: Solving and Graphing Linear Inequalities

Slide 115 / 182

Shading is above a solid line.This means

the solutions are above the line AND on it.

Shading is below a solid line. This means the solutions are below

the line AND on it.

The following are graphs of linear inequalities.Graphing

Page 116: Solving and Graphing Linear Inequalities

Slide 116 / 182

How to Graph a Linear Inequality

1) Decide where the boundary goes: Solve inequality for y, for example y > 2x - 1

2) Decide whether boundary should be: - solid (≤ or ≥: points on the boundary make the inequality true) or - dashed (< or >: points on the boundary make the inequality false)

3) Graph the boundary (the line).

4) Decide where to shade: y > or y ≥: shade above (referring to y-axis) the boundary y < or y ≤: shade below (referring to y-axis) the boundary Or, you can test a point

Page 117: Solving and Graphing Linear Inequalities

Slide 117 / 182

Graph

Step 1: Solve for y: (Think ), m = -2 and b = 1

Step 2: The line should be dashed because the inequality is <

Step 3: Graph boundary

Step 4: Shade below the boundary line because y <

Graphing

Page 118: Solving and Graphing Linear Inequalities

Slide 118 / 182

Graph

Step 1: Solve for y

Step 2: The line should be solid because the inequality is ≥

Step 3: Graph boundary

Step 4: Shade above the boundary line because y ≥

Graphing

Page 119: Solving and Graphing Linear Inequalities

Slide 119 / 182

Graph

Is the equation already solved for y?

Is the line solid or dashed? Explain why this is the case.The line is dashed because it is not included in the inequality.

Will we shade above or belowthe line? Explain why this is thecase.You shade above the line because the inequality showsthat y is greater than the expression on the right hand side. Or, if you test a point (0, 0),it satisfies the inequality, so you shade in that direction.

Graphing

click to reveal

click to revealclick to reveal the inequality graph

Page 120: Solving and Graphing Linear Inequalities

Slide 120 / 182

51 Why are there dashed boundaries on some graphs of inequalities?

A Points on the line make the inequality false.B Points on the line make the inequality true.C The slope of the line depends on the line type.D The y-intercept depends on the line type.

Page 121: Solving and Graphing Linear Inequalities

Slide 121 / 182

52 For which of these inequalities would the graph have a solid boundary and be shaded above?

ABC

D

Page 122: Solving and Graphing Linear Inequalities

Slide 122 / 182

53 For which of these inequalities would the graph have a dashed boundary and be shaded above?

ABC

D

Page 123: Solving and Graphing Linear Inequalities

Slide 123 / 182

54 Which inequality is graphed?

A

B

C

D

Page 124: Solving and Graphing Linear Inequalities

Slide 124 / 182

Page 125: Solving and Graphing Linear Inequalities

Slide 125 / 182

56 Graph the solution set of . When you finish, type the number "1" into your responder.

PARCC - EOY - Question #2 Non-Calculator Section - SMART Response Format

Page 126: Solving and Graphing Linear Inequalities

Slide 126 / 182

Modeling with Inequalities

Throughout this unit, you have learned how to solve and graph inequalities, both on a number line and in the coordinate plane.

We can apply these skills to solve realistic word problems, such as purchasing items at a store within a budget and earning money through various jobs.

Let's get started.

Page 127: Solving and Graphing Linear Inequalities

Slide 127 / 182

Modeling with Inequalities

At a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 each. You have $125 to spend. Let x represents the dress shirts and y represents the number of pairs of dress pants.

Part AWrite an inequality that would be used to model the situation.

Part BGraph the inequality in a coordinate plane.

Part CList 3 combinations of dress shirts and pairs of dress pants that could be purchased within your budget.

Page 128: Solving and Graphing Linear Inequalities

Slide 128 / 182

Modeling with Inequalities

At a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 each. You have $125 to spend. Let x represents the dress shirts and y represents the number of pairs of dress pants.

Part AWrite an inequality that would be used to model the situation.

Page 129: Solving and Graphing Linear Inequalities

Slide 129 / 182

15

20

15 20

5

5x

10

0 10

y

Modeling with InequalitiesAt a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 each. You have $125 to spend. Let x represents the dress shirts and y represents the number of pairs of dress pants.Part BGraph the inequality in a coordinate plane.

Page 130: Solving and Graphing Linear Inequalities

Slide 130 / 182

Modeling with Inequalities

At a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 each. You have $125 to spend. Let x represents the dress shirts and y represents the number of pairs of dress pants.

Part CList 3 combinations of dress shirts and pairs of dress pants that could be purchased within your budget.

Page 131: Solving and Graphing Linear Inequalities

Slide 131 / 182

57 At a sports shop, soccer balls cost $18 each and footballs cost $15 each. You have $90 to spend. Let x represents the number of soccer balls and y represents the number of footballs.

Part AWhich inequality would be used to model this situation?

A

B

C

D

Page 132: Solving and Graphing Linear Inequalities

Slide 132 / 182

58 At a sports shop, soccer balls cost $18 each and footballs cost $15 each. You have $90 to spend. Let x represents the number of soccer balls and y represents the number of footballs.

15

20

15 20

5

5x

10

0 10

yPart BGraph your solution in the coordinate plane below. When you are finished, type the number "1" into your responder.

Page 133: Solving and Graphing Linear Inequalities

Slide 133 / 182

59 At a sports shop, soccer balls cost $18 each and footballs cost $15 each. You have $90 to spend. Let x represents the number of soccer balls and y represents the number of footballs.

Part CWhich pairs (x, y) can represent the amount of soccer balls and footballs purchased at the sports shop? Select all that apply.

A (7, 1)

B (2, 3)

C (4, 6)

D (3, 3)

E (1, 4)

Page 134: Solving and Graphing Linear Inequalities

Slide 134 / 182

60 A group of friends went to the movies on Friday night. After purchasing the tickets, they had $30 left to spend on soda, which costs $1.50 per cup and popcorn, which costs $4.50 per bucket. Let x represent the number of sodas purchased and y represent the buckets of popcorn purchased.

Part AWhich inequality would be used to model this situation?

A

B

C

D

Page 135: Solving and Graphing Linear Inequalities

Slide 135 / 182

61 A group of friends went to the movies on Friday night. After purchasing the tickets, they had $30 left to spend on soda, which costs $1.50 per cup and popcorn, which costs $4.50 per bucket.

15

20

15 20

5

5x

10

0 10

yLet x represent the number of sodas purchased and y represent the buckets of popcorn purchased.

Part BGraph your solution in the coordinate plane below. When you are finished, type the number "1" into your responder.

Page 136: Solving and Graphing Linear Inequalities

Slide 136 / 182

62 A group of friends went to the movies on Friday night. After purchasing the tickets, they had $30 left to spend on soda, which costs $1.50 per cup and popcorn, which costs $4.50 per bucket. Let x represent the number of sodas purchased and y represent the buckets of popcorn purchased.

Part CWhich pairs (x, y) can represent the amount spent on soda and buckets of popcorn at the theater? Select all that apply.

A (17, 1)

B (10, 5)

C (8, 4)

D (5, 5)

E (3, 7)

Page 137: Solving and Graphing Linear Inequalities

Slide 137 / 182

Solving Systemsof Inequalities

Return to Tableof Contents

Page 138: Solving and Graphing Linear Inequalities

Slide 138 / 182

Vocabulary

A system of linear inequalities is two or more linear inequalities.

The solution to a system of linear inequalities is the intersection of the half-planes formed by each linear inequality.

The most direct way to find the solution to a system of linear inequalities is to graph the equations on the same coordinate plane and find the region of intersection.

Page 139: Solving and Graphing Linear Inequalities

Slide 139 / 182

Step 1: Graph the boundary lines of each inequality.

Remember: - dashed line for < and > - solid line for ≤ and ≥

Step 2: Shade the half-plane for each inequality.

Step 3: Identify the intersection of the half-planes. This is the solution to the system of linear inequalities.

Graphing a System of Linear Inequalities

Page 140: Solving and Graphing Linear Inequalities

Slide 140 / 182

Solve the following system of linear inequalities.

Step 1:

Example

5

10

5 10

-5

-5x

-10

0-10

y

Page 141: Solving and Graphing Linear Inequalities

Slide 141 / 182

5

10

5 10

-5

-5x

-10

0-10

y

Example Continued

Step 2:

Page 142: Solving and Graphing Linear Inequalities

Slide 142 / 182

5

10

5 10

-5

-5x

-10

0-10

y

Example Continued

Step 3 :

Page 143: Solving and Graphing Linear Inequalities

Slide 143 / 182

Solve the following system of linear inequalities.

Step 1:

Example

5

10

5 10

-5

-5x

-10

0-10

y

Page 144: Solving and Graphing Linear Inequalities

Slide 144 / 182

5

10

5 10

-5

-5x

-10

0-10

y

Example Continued

Step 2:

Page 145: Solving and Graphing Linear Inequalities

Slide 145 / 182

Example Continued

Step 3:

5

10

5 10

-5

-5x

-10

0-10

y

Page 146: Solving and Graphing Linear Inequalities

Slide 146 / 182

Solve the following system of linear inequalities.Example

5

10

5 10

-5

-5x

-10

0-10

yStep 1:

Page 147: Solving and Graphing Linear Inequalities

Slide 147 / 182

Example Continued

Step 2:

5

10

5 10

-5

-5x

-10

0-10

y

Page 148: Solving and Graphing Linear Inequalities

Slide 148 / 182

Example Continued

Step 3:

5

10

5 10

-5

-5x

-10

0-10

y

Page 149: Solving and Graphing Linear Inequalities

Slide 149 / 182

Page 150: Solving and Graphing Linear Inequalities

Slide 150 / 182

Page 151: Solving and Graphing Linear Inequalities

Slide 151 / 182

Page 152: Solving and Graphing Linear Inequalities

Slide 152 / 182

63 Choose the graph below that displays the solution to the following system of linear inequalities:

A B C

Page 153: Solving and Graphing Linear Inequalities

Slide 153 / 182

Page 154: Solving and Graphing Linear Inequalities

Slide 154 / 182

65 Choose the graph below that displays the solution to the following system of linear inequalities:

A B C

Page 155: Solving and Graphing Linear Inequalities

Slide 155 / 182

66 Choose the graph below that displays the solution to the following system of linear inequalities:

A B C

Page 156: Solving and Graphing Linear Inequalities

Slide 156 / 182

67 Choose all of the linear inequalities that correspond to the following graph:

A

B

C

D

Page 157: Solving and Graphing Linear Inequalities

Slide 157 / 182

68 Which point is in the solution set of the system of

inequalities shown in the accompanying graph?

A (0, 4)

B (2, 4)

C (-4, 1)

D (4, -1)From the New York State Education Department. Office of Assessment Policy, Development and

Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 158: Solving and Graphing Linear Inequalities

Slide 158 / 182

69 Which ordered pair is in the solution set of the system of inequalities shown in the accompanying graph?

A (0, 0)

B (0, 1)C (1, 5)D (3, 2)

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 159: Solving and Graphing Linear Inequalities

Slide 159 / 182

70 Which ordered pair is in the solution set of the following system of linear inequalities?

A (0, 3)

B (2, 0)

C (−1, 0)

D (−1, −4)

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 160: Solving and Graphing Linear Inequalities

Slide 160 / 182

71 Mr. Braun has $75.00 to spend on pizzas and soda for a picnic. Pizzas cost $9.00 each and the drinks cost $0.75 each. Five times as many drinks as pizzas are needed. What is the maximum number of pizzas that Mr. Braun can buy?

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Page 161: Solving and Graphing Linear Inequalities

Slide 161 / 182

72 A system of inequalities is given.

PARCC - PBA - Question #3 Non-Calculator Section - SMART Response Format

Graph the solution set of the system of linear inequalities in the coordinate plane.When you finish, type the number "1" into your Responder.

Page 162: Solving and Graphing Linear Inequalities

Slide 162 / 182

Modeling with a System of Inequalities

Similar to solving application problems by graphing a single inequality, we can also apply our skills with solving a system of inequalities to solve realistic word problems.

Let's get started.

Page 163: Solving and Graphing Linear Inequalities

Slide 163 / 182

Modeling with a System of InequalitiesPreston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli.

Part A: Graph the solution set of the system of linear inequalities in a coordinate plane.Part B: Create 3 ordered pairs (x, y) that represent the hours that Preston could work to meet the given conditions.Part C: Given the conditions in Part A, if Preston mows lawns for 9 hours this month, what is the minimum number of hours he would have to work at the deli to earn at least $150? Give your answer to the nearest whole hour.Part D: Given the conditions in Part A, Preston prefers mowing lawns over working at the deli. What is the maximum number of hours he can mow lawns to be able to earn at least $150? Give your answer to the nearest whole hour.

Page 164: Solving and Graphing Linear Inequalities

Slide 164 / 182

Modeling with a System of InequalitiesPreston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot

15

20

15 20

5

5x

10

0 10

ywork more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli.Part A: Graph the solution set of the system of linear inequalities in a coordinate plane.

Page 165: Solving and Graphing Linear Inequalities

Slide 165 / 182

Modeling with a System of InequalitiesPreston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli.

Part B: Create 3 ordered pairs (x, y) that represent the hours that Preston could work to meet the given conditions.

Page 166: Solving and Graphing Linear Inequalities

Slide 166 / 182

Modeling with a System of InequalitiesPreston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli.

Part C: Given the conditions in Part A, if Preston mows lawns for 5 hours this month, what is the minimum number of hours he would have to work at the deli to earn at least $150? Give your answer to the nearest whole hour.

Page 167: Solving and Graphing Linear Inequalities

Slide 167 / 182

Modeling with a System of InequalitiesPreston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli.

Part D: Given the conditions in Part A, Preston prefers mowing lawns over working at the deli. What is the maximum number of hours he can mow lawns to be able to earn at least $150? Give your answer to the nearest whole hour.

Page 168: Solving and Graphing Linear Inequalities

Slide 168 / 182

73 Gavin is selling comic books and baseball cards to make money for summer vacations. The comic books each cost $6 and baseball cards cost $5 for a single pack. He needs to make at

30

40

30 40

10

10x

20

0 20

yleast $210. Gavin knows that the will sell more than 20 comic books. Let x represent the number of comic books sold and y represent the packs of baseball cards sold.

Part A: Graph the solution set of the system of linear inequalities in a coordinate plane. When you finish, type the number "1" into your Responder.

Page 169: Solving and Graphing Linear Inequalities

Slide 169 / 182

74 Gavin is selling comic books and baseball cards to make money for summer vacations. The comic books each cost $6 and baseball cards cost $5 for a single pack. He needs to make at least $210. Gavin knows that the will sell more than 20 comic books. Let x represent the number of comic books sold and y represent the packs of baseball cards sold.

Part BWhich pairs (x, y) represent the sales of comic books and packs of baseball cards to meet the given conditions? Select all that apply.

A (25, 25)

B (26, 8)

C (30, 10)

D (35, 25)

E (18, 40)

Page 170: Solving and Graphing Linear Inequalities

Slide 170 / 182

75 Gavin is selling comic books and baseball cards to make money for summer vacations. The comic books each cost $6 and baseball cards cost $5 for a single pack. He needs to make at least $210. Gavin knows that the will sell more than 20 comic books. Let x represent the number of comic books sold and y represent the packs of baseball cards sold.

Part CGiven the conditions in Part A, if Gavin sold 14 packs of baseball cards, what is the minimum number of comic books he would need to sell to earn at least $210? Give your answer to the nearest whole number.

Page 171: Solving and Graphing Linear Inequalities

Slide 171 / 182

76 Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop.

PARCC - EOY - Question #25 Calculator Section - SMART Response Format

Part AGraph the solution set of the system of linear inequalities in the coordinate plane.When you finish, type the number "1" into your Responder.

Page 172: Solving and Graphing Linear Inequalities

Slide 172 / 182

77 Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop.

Part BWhich pairs (x, y) represent hours that Leah could work to meet the given conditions? Select all that apply.

A (4, 15)

B (5, 12)

C (10, 9)

D (15, 5)

E (19, 1)

PARCC - EOY - Question #25 Calculator Section

Page 173: Solving and Graphing Linear Inequalities

Slide 173 / 182

78 Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop.

Part CGiven the conditions in Part A, if Leah babysits for 7 hours this month, what is the minimum number of hours she would have to work at the ice cream shop to earn at least $120? Give your answer to the nearest whole hour.

PARCC - EOY - Question #25 Calculator Section

Page 174: Solving and Graphing Linear Inequalities

Slide 174 / 182

79 Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop.

Part DGiven the conditions in Part A, Leah prefers babysitting over working at the ice cream store. What is the maximum number of hours she can babysit to be able to earn at least $120? Give your answer to the nearest whole hour.

PARCC - EOY - Question #25 Calculator Section

Page 175: Solving and Graphing Linear Inequalities

Slide 175 / 182

Glossary & Standards

Return toTable ofContents

Page 176: Solving and Graphing Linear Inequalities

Slide 176 / 182

Back to

Instruction

InequalityAn Inequality is a mathematical sentence that uses symbols, such as <, ≤, > or ≥ to compare

to quantities.

x > 6

x ≤ -3

2 < 18r ≥ 11

+ 9 +9r - 9 ≥ 2

Page 177: Solving and Graphing Linear Inequalities

Slide 177 / 182

r ≥ 11

+ 9 +9r - 9 ≥ 2

Back to

Instruction

Solution is included!

Solution is not included!

Solution SetAny number that, when substituted into an equation/inequality, will satisfy the equation/

inequality

r - 9 = 2+ 9 +9

r = 11

check:11 - 9 = 2

2 = 2

{11}

Page 178: Solving and Graphing Linear Inequalities

Slide 178 / 182

"and" means intersection

"or" means union

x > -2 AND x < 3 -2 < x < 3

x ≤ -2 OR x ≥ 3

Back to

Instruction

Compound Inequality

Two inequalities that are combined into one statement by the words AND/OR

Page 179: Solving and Graphing Linear Inequalities

Slide 179 / 182

Back to

Instruction

2x + 8 = 2(x - 4)

2x + 8 = 2x - 8

8 = -8

{ } or ∅

2x ≥ 18 AND -3x > 12

x ≥ 9 AND x < -4

No Solution

When there is no number that makes the equation/inequalities true

{ }

"no solution"∅

Page 180: Solving and Graphing Linear Inequalities

Slide 180 / 182

Back to

Instruction

-2x + 3 > 17 OR

5(x + 2) > -40

x ≤ -7 OR x > -10

R

Reals

"all real numbers"

"reals"

R

When all (any) numbers make the equation/inequalities true

2x + 8 = 2(x + 4)

2x + 8 = 2x + 8

R0 = 0

Page 181: Solving and Graphing Linear Inequalities

Slide 181 / 182

Back to

Instruction

System of Linear InequalitiesTwo or more linear inequalities

y > 2x - 3y < -x + 4

5

10

5 10

-5

-5x

-10

0-10

y

Page 182: Solving and Graphing Linear Inequalities

Slide 182 / 182

Throughout this unit, the Standards for Mathematical Practice are used.

MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for and express regularity in repeated reasoning.

Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.

If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.