Name ___________________________________________ Grade________________________ Teacher__________________________________________

Inequalities

< >

Graphing and Writing Inequalities Classwork Day 1 Word Map:

> <

Greater than

Greater than or equal to

At Least

Minimum

Less than Less than or equal to

At Most

Maximum

When graphing use:

When graphing use:

When graphing use:

When graphing use:

Translate, graph, and write the solution set for each inequality.

1) x > 3 2) 3x 3) -4 > x 4) 1 < x

__________________ __________________ __________________ __________________

Graphing Inequalities

We can represent a linear inequality in one variable on a number line.

We use the following symbols in the representation:

An ________________ circle is used for < and > to indicate that the number is not included.

A _________________ or filled-in circle is used for and to indicate that the number is included.

A line with an _______________ indicates that the line continues to ___________________ in the

direction of the arrow.

Guided Practice

Graph the following inequalities and list the solution set:

1) x 3 2) x < 5 3) 7 x 4) -2 > x

Graphing and Writing Inequalities Classwork Day 1

Write the inequality to represent the graph.

5) 6) 7) 8)

2 -5 0 8

ans. ____________ ans. ____________ ans. ____________ ans. ____________

Independent Practice

1) Graph the following inequalities and list the solution set:

a) x 9 b) x < -7 c) 0 x d) -1 > x

e) x < -10 f) x -3 g) 11 > x h) x -13

2) Write the inequality to represent the graph.

a) b) c) d)

8 6 -2 4

ans. ____________ ans. ____________ ans. ____________ ans. ____________

3) Translate and graph.

Sentence Inequality Graph

x is greater than 3

You must be at least 16 to drive.

x is less than or equal to 3

Graphing and Writing Inequalities Classwork Day 1 Translate and write the solution set for each inequality.

1) x > 5 2) 9x 3) x < 8 4) x6

________________ ________________ ________________ ________________

5) Graph the following inequalities and list the solution set:

e) x > -6 f) x -4 g) 10 > x h) x -3

Write the inequality to represent the graph.

6) 7) 8) 9)

-8 2 12 7

ans. ____________ ans. ____________ ans. ____________ ans. ____________

Write an inequality to represent a situation.

10) Florencia has $60 to spend on clothes. She wants to buy a pair of jeans for $22 dollars and spend the rest

on t-shirts. Each t-shirt costs $8. Write an inequality for the number of t-shirts she can purchase.

11) As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at more

than $100. Write an inequality for the number of sales you need to make, and describe the solutions.

12) A local car dealership is trying to sell all of the cars that are on the lot. Currently, it has 525 cars on the lot,

and the general manager estimates that they will consistently sell 50 cars per week. Estimate how many weeks

it will take for the number of cars on the lot to be less than 75.

Write an inequality that can be used to find the number of 𝑤 full weeks. Since 𝑤 is the number of full or

complete weeks, when 𝑤 = 1 means at the end of week 1.

Solve Inequalities 1-2 Steps With Positive Coefficients Classwork Day 2

To solve inequalities, follow the same processes that apply to equations.

►Example

Graph the solution set for 5x 4 < 16

+ +

5x <

x <

Graph Solution set

Try These.

Directions: Solve, graph, and list the solution set of the given inequality.

1. y + 3 4

Solution set

2. 2x 4 > 12

Solution set

3. 3n + 4 2

Solution set

4. 20 > 5x 15

Solution set

5. 310

t

Solution set

6. 17 8y + 1

Solution set

7. 54 > 6h

Solution set

8. 1024

t

Solution set

9. x 2 80

Solution set

10. 6y + 5 > 11

Solution set

11. 821 d

Solution set

12. 12 < 8 + 5b

Solution set

Real-World Inequalities

13) At most, Kyle can spend $50 on sandwiches and chips for a picnic. He already bought chips for $6 and will

buy sandwiches that cost $4.50 each.

14) A youth summer camp has budgeted $2000 for the campers to attend the carnival. The cost for each

camper is $17.95, which includes general admission to the carnival and 2 meals. The youth summer camp must

also pay $250 for the chaperones to attend the carnival and $350 for transportation to and from the carnival.

What is the greatest amount of campers that can attend the carnival if the camp must stay within their budgeted

amount?

15) Cathy has $100 saved to spend on clothes. She wants to purchase a winter jacket for $40 and some

sweaters that cost $20 each. Write, solve and graph an inequality to show how many sweater she can buy.

Show your work and interpret your solution.

Think* Is it possible to have negative sweaters? Graph the individual points.

CONCLUSION

Why do we graph our solution of an inequality on a number line and we didn’t graph our answer in an

equation? (Explain in words)

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

4321012

Solve Inequalities 1-2 Steps With Positive Coefficients Classwork Day 2

Directions: Solve, graph, and list the solution set for each inequality.

1. 7x 3 25

Solution set

2. 3h + 11 2

Solution set

3. 6 + 2q < 2

Solution set

4. 5 + 6k 11

Solution set

5. 3x < 12

Solution set

6. 2

y

+ 2 < 8

Solution set

Directions: Write an inequality for each graph below.

7.

8.

9. Kate took $3 out of her purse, and she still had at least $8 in it. How much did she have to begin?

Part A Write the inequality ___________________ Part B Solve the inequality.

10) The carnival owner pays the owner of an exotic animal exhibit $650 for the entire time the exhibit is

displayed. The owner of the exhibit has no other expenses except for a daily insurance cost. If the owner of the

animal exhibit wants to make more than $500 in profits for the 512 days, what is the greatest daily insurance

rate he can afford to pay?

Solve 1-2 Step Inequalities With Integers (Negative Coefficients) Classwork Day 3 Warm-up

Given that 3 < 6, would this inequality be preserved (Stay in the same direction) or reversed if we:

a. Added 2 to both sides

b. Subtracted 3 from both sides

c. Multiplied both sides by .5

d. Divided both sides by 3

e. Multiplied both sides by -4

f. Divided both sides by -3

Explore

Inequality Multiply each side by: New Inequality New Inequality is True or False?

3 < 4 2

2 -3 3

-1 6 5

5 > 2 -1

1 7 -5

-8 > -10 -8

Inequality Divide each side by: New Inequality New Inequality is True or False?

4 < 8 4 12 -15 3 -16 12 -4

15 > 5 -5

►Example

Solve the inequalities: 3c < 60 3c < 60 3c < -60

Use substitution to ensure the solution makes the inequality true.

◊ TIP: If 0 is a value that satisfies your answer, it can be a useful number to substitute when

checking your solution.

◊Rule: The direction of the inequality symbol changes only when multiplying or dividing by a

negative number.

Directions: Solve and graph the inequality on the given number line.

1. y + 9 14

Solution set

2. y + 9 14

Solution set

3. 2 3n + 1

Solution set

4. 24 < 8x

Solution set

5. 65

t

Solution set

6. 24 6y

Solution set

7. 14 > 7h

Solution set

8. 33

t

Solution set

9. x 8 8

Solution set

10. y + 7 > 13

Solution set

11. 216 d

Solution set

12. 10 < 6 + x

Solution set

When both sides of an inequality are multiplied or divided by a ___________number, the inequality is no

longer true.

Inequality Multiply each side by: New Inequality Reverse the

inequality symbol

Reversed symbol makes

it True or False?

5 > 2 -1

1 7 -5

-8 > -10 -8

Inequality Divide each side by: New Inequality Reverse the

inequality symbol

Reversed symbol makes

it True or False?

-16 12 -4

15 > 5 -5

Conjecture- When both sides of an inequality are multiplied or divided by a negative number; you must

_________________ to make the statement true.

Solve 1-2 Step Inequalities With Integers (Negative Coefficients) Classwork Day 3 Directions: Solve, graph, and list the solution set for each inequality.

1. x 7 14

Solution set

2. 2x + 8 20

Solution set

3. 8 + 3q < 4

Solution set

4. 6 2k 10

Solution set

5. 9x < 36

Solution set

6. 2

y+ 3 > 8

Solution set

Determine if the following is an equation, expression, or inequality.

7. x = 6 8. 2x + y2 9. x > 7 10. 2x 10

11. For numbers 7– 10, which one is a polynomial? Justify your answer.

Equation review. Solve the following.

12. 10 = 4(3 + x) 13. 40(x – 2) = 200 14. 242

8

x

15. For each problem, use the properties of inequalities to write a true inequality statement. Two integers are

−2 and −5.

a. Write a true inequality statement.

b. Subtract −2 from each side of the inequality. Write a true inequality statement.

c. Multiply each number by −3. Write a true inequality statement.

Solve 1 & 2 step inequalities with rational numbers Classwork Day 4 Warm Up

1) What does the symbol mean? ___________________________________________

2) Is 0 + 2 less than or equal to 5? 3) Is 1 + 2 less than or equal to 5?

4) Is 2 + 2 less than or equal to 5? 5) Is 3 + 2 less than or equal to 5?

6) Is 4 + 2 less than or equal to 5? 7) Is 1 (4) less than or equal to 5?

►Example

Solve and Graph the inequality:

123

2x

1. 1022

1x

2. 20

12

4

3 x

3. 16 0.25x

4. 102

12 x

5. 2.4x 5 < 25

6. 64

21

8

7x

◊ Think…

*What is the coefficient?

*If the coefficient were 3, what would be the first step in

solving the inequality?

*Using that strategy, what is the first step in solving

123

2x ?

*Dividing by 3

2 is the same as multiplying by what number?

Solve 1 & 2 step inequalities with rational numbers Classwork Day 4

Try These.

Solve and graph the following inequalities.

1) 15115

x 2) 2.8p 4.2 3) 140 x

15

7

4) 15 + 353

13 x 5) 7 < 1 + x

3

2 6) 27 + 8.5n 78

7) 2

13

3

242 x 8) 2.5p + 3.9 > 8.1

Ticket out the door (separate piece of paper) or one more for the road

Solve the following inequality and graph the solution.

4712

1125 m

Solve 1 & 2 step inequalities with rational numbers Classwork Day 4 Solve and graph the following inequalities.

1) 2

141

2

x 2) 2x – 3.1 1.7 3) 10

3

22 x

4) 6x + 14 20 5) 4313

x

6) 5x – 11 38

Write, solve and graph an inequality for each sentence.

7) Three times a number increased by four is less than 62

8) The quotient of a number and 5 increased by one is more than 7.

9) The product of 2 and a number minus six is greater than 18.

10) The quotient of a number and three minus two is no more than 12.

11) What is the solution to 145

3 x ?

A x > -5 B x < -5 C x > 5 D x < 5

Translating Inequalities Classwork Day 5

An inequality is a mathematical sentence that compares two quantities that are not equal. Use the following

symbols to represent inequalities:

< means “is less than.” means “is less than or equal to”

> means “is greater than.” means “is greater than or equal to”

Like equations, you can write inequalities to represent a situation.

a) How could you represent, Lisa will spend less than $25?

b) How could you represent, Rodney ran at least 30 miles last week?

c) How could you use a number line to show greater than 2?

Try These

Write an inequality to represent the given situation.

Sentence Inequality

1) A number decreased by five is less than fourteen.

2) Five times a number is at least twenty.

3) The difference between half a number and four is at most ten.

4) The sum of four times a number and six is more than twelve.

5) The quotient of 6 and n is greater than or equal to 8.

Practice

Sentence Inequality

6) The sum of four and n is greater than fifteen.

7) A number increased by two is less than or equal to four.

8) You must be at least forty inches tall to go on the ride.

9) Tom wants to spent at most $50 for a pair of sneakers.

10) Twice a number x is less than one-third.

Using the above examples, what was the inequality used for:

At Most _____________ At Least__________________

(maximum) (minimum)

Translating Inequalities Classwork Day 5 1) On a particular airline, checked bags can weigh no more than 50 pounds. Sally packed 32 pounds of

clothes and five identical gifts in a suitcase that weigh 8 pounds. Write an inequality to represent this

situation.

2) Match each problem to the inequality that models it. One choice will be used twice.

____ The sum of three times a number and 4 is greater than 17. a. 3𝑥+ 4 ≥ 17

____ The sum of three times a number and 4 is less than 17. b. 3𝑥+ 4 < 17

____ The sum of three times a number and 4 is at most 17. c. 3𝑥+ 4 > 17

_____The sum of three times a number and 4 is no more than 17. d. 3𝑥+ 4 ≤ 17

____ The sum of three times a number and 4 is at least 17.

3) The length of a rectangular fenced enclosure is 12 feet more than the width. If Farmer Dan has 100 feet

of fencing, write an inequality to find the dimensions of the rectangle with the largest perimeter that can

be created using 100 feet of fencing.

4) A group of up to 40 people are going on a trip to Washington, DC. Some will travel in a van that holds

12 people, and the rest will buy train tickets. Write an inequality that can be used to find the number of

train tickets that the group will need.

5) A used bookstore is having a sale. All paperback books are $0.75 and all hardcover books are $3.00.

Ginny has $10. She wants to buy one hardcover book and as many paperback books as she can. Which

inequality represents this situation, where p represents the number of paperback books Ginny can buy?

A. 3 + 0.75p < 10 B. 3 + 0.75p 10

C. 3 + 0.75p 10 D. 3 + 0.75p > 10

6) Caitlyn is shopping online to find a new television for the center. Caitlyn wants a television with at

least a 26-in. screen.

Television Prices

Screen Size Price

22 in $300

26 in $330

32 in $370

40 in $420

Write an inequality to show how much money, m, the center will need to spend.

Translating Inequalities Classwork Day 5 Write an inequality to represent the given situation.

1) A number x multiplied by two, decreased by four, is greater than nine.

2) Seven tenths of a number x is at most forty-nine.

3) Five times a number x, increased by six, is less than or equal to negative twenty.

4) The quotient of a number and four, decreased by 8

3, is at least

8

1 .

5) The cost to run an ad in a newspaper is $10 plus $0.25 per word. Write an inequality that can be used to

find the maximum number of words Marianne can put in her ad if the most she can spend is $15.00.

Amusement Park Rides Height Requirements

Ride Height

Requirements

Ride Height

Requirements

Jungle Jam minimum of 40 in. The Spiral more than 35 in.

Tilt-a-Whirl at least 48 in. Ladybug under 46 in.

Stargazer more than 44 in. Leapin’ Lizard at least 38 in.

Bunny Hop 60 in. maximum Racetrack over 42 in.

6. Using the table above, write and graph an inequality that represents the heights of people who can ride the

Tilt-a-Whirl.

7. Write and graph an inequality that represents the heights of people who can ride the Ladybug.

8. Is it possible that someone is able to ride the Spiral but not the Leapin’ Lizard? If so, give that person’s

height.

9. Would someone be able to ride both the Tilt-a-Whirl and the Ladybug? Yes or No

Explain your answer. ___________________________________________________________

____________________________________________________________________________

____________________________________________________________________________

Application Problems Classwork Day 6 Warm Up

1) Part A: Given the inequality: 2x + 4 12 Make up your own word problem to represent the inequality.

Part B: Solve the inequality. Part C: Graph the inequality.

1) A rental company charges $45 plus $0.20 per mile to rent a car. Mr. DeMeo does not want to spend more

than $100 for his rental car. Write, solve and graph an inequality to find how many miles he can drive and not

spend more than $100. Interpret the solution.

Inequality __________________________ Solve the inequality

Graph the solution

2) Halfway through the bowling league season, Nick has 34 strikes. He averages 2 strikes per game. Write,

solve and graph an inequality to find how many more games it will take for Nick to have at least 61 strikes, the

league record.

Inequality __________________________ Solve the inequality

Graph the solution

3) Tyler needs at least $205 for a new video game system. He has already saved $30. He earns $7 an hour at

his job. Write, solve and graph an inequality to find how many hours he will need to work to buy the system.

Inequality __________________________ Solve the inequality

Graph the solution

4) A plate weighs 4

1 pound. A shelf can hold at most 20 pounds. Write, solve and graph an inequality to find

how many plates the shelf can hold.

Inequality __________________________ Solve the inequality

Graph the solution

Application Problems Classwork Day 6 1) Katie is starting a babysitting business. She spent $26 to make signs to advertise. She charges an initial fee

of $5 and then $3 for each hour of service. Write, solve and graph an inequality to find the number of hours

she will have to babysit to make a profit.

Inequality __________________________ Solve the inequality

Graph the solution

2) As a salesperson, Audrey earns $75 per week plus $5 per sale. This week, she wants to pay to be at least

$125. Write, solve and graph an inequality for the number of sales Audrey needs to make.

Inequality __________________________ Solve the inequality

Graph the solution

3) Dylan and his sister went to the movies. They had $34 altogether and spent $9.50 per ticket. Dylan and his

sister bought the same snacks. Write, solve and graph an inequality for the amount that each person spent on

snacks.

Inequality __________________________ Solve the inequality

Graph the solution

M3L15 4) Mrs. Smith decides to buy three sweaters and a pair of jeans. She has $120 in her wallet. If the price of the jeans is $35, what is the highest possible price of a sweater?

Inequality __________________________ Solve the inequality

Graph the solution

Mixed Review

Simplify the following expressions.

5) (22 + 19b) + 7 6) 10y(7) 7) 8)6(3

1x 8) (12 + 4)3

9) 7x – 2 – 7x + 6 10) 9y – 4 – 11y + 7 11) 9 – m + 3 – 2m