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Algebra I

Solving & Graphing Inequalities

2016-01-11

www.njctl.org

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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

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Simple Inequalities Involving Additionand Subtraction

Return to Table of Contents

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

Inequality

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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)

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http://www.njctl.org

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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

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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

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· 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

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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

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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

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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/

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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?

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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 ?

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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



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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


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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


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Simple Inequalities Involving Multiplication

and Division


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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

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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

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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:


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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

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Find the solution to the inequality.9

A

B

C

D

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10

A

B

C

Find the solution to the inequality.

D


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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

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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

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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

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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.

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13

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


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14

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


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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

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Solving Two-Step and Multiple-Step

InequalitiesReturn to Table of Contents

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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

A 8B 9C 12D 16


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23 What is the solution of ?

ABCD


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

ABCD


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

A 18

B 6

C -3D -12


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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.

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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

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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.


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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?


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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?


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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.


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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


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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


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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

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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

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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

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Solving CompoundInequalities


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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.

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

31 Which inequality is represented in the graph below?

AB

C

D


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

32 Which inequality is represented in the graph below?

A

B

C

D


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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

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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

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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

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A

B

C

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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

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A

B

C

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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

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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

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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

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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

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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


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

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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


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41 Solve

A

B

C

D

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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?

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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.


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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.


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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

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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

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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

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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

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Special Cases of Compound Inequalities


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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.

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Solve each set of compound inequalities.

1. and

2. or

Special Cases

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Solve each set of compound inequalities.

3. and

4. and

Special Cases

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Graphing Linear Inequalitiesin Slope-Intercept Form


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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

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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

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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

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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

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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

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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

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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.

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

ABC

D

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

ABC

D

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54 Which inequality is graphed?

A

B

C

D

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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

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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.

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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.

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

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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.

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

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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

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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.

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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)

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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

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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.

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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)

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Solving Systemsof Inequalities

Return to Tableof Contents

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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.

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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

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Solve the following system of linear inequalities.

Step 1:

Example

5

10

5 10

-5

-5x

-10

0-10

y

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5

10

5 10

-5

-5x

-10

0-10

y

Example Continued

Step 2:

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5

10

5 10

-5

-5x

-10

0-10

y

Example Continued

Step 3 :

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Solve the following system of linear inequalities.

Step 1:

Example

5

10

5 10

-5

-5x

-10

0-10

y

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5

10

5 10

-5

-5x

-10

0-10

y

Example Continued

Step 2:

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Example Continued

Step 3:

5

10

5 10

-5

-5x

-10

0-10

y

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Solve the following system of linear inequalities.Example

5

10

5 10

-5

-5x

-10

0-10

yStep 1:

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Example Continued

Step 2:

5

10

5 10

-5

-5x

-10

0-10

y

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Example Continued

Step 3:

5

10

5 10

-5

-5x

-10

0-10

y

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

A B C

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A B C

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A B C

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

A

B

C

D

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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.

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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)


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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)


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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?


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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.

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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.

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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.

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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.

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

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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.

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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.

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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.

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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)

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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.

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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.

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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

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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.


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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.


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Glossary & Standards

Return toTable ofContents

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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

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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}

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"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

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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"∅

∅

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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

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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

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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.

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