Name: ______________________________
Unit 5: Solving Inequalities
Integrated Algebra 1A
Day 1 –Inequalities & Their Graphs
What is an Inequality?
An inequality is a statement that shows the relationship between two (or more)
expressions with one of the following signs:  .
x < y means "
x ≤ y means "
x > y means "
x ≥ y means "
x is
x is
x is
x is
__________________________________________
__________________________________________
__________________________________________
__________________________________________
y"
to y "
than y "
to y "
Write an inequality that represents each verbal expression.
1. v is greater than 10
2. b is less than or equal to 1
____________________________
____________________________
3. the product of g and 2 is less than or equal to 6 __________________________
4. 2 more than k is greater than 3
___________________________
1
Summary of symbols: Multiple Choice, choose the appropriate symbol for each set of
words.

Is at least: __________
(a) 
(b) 
(c) 
(d) 

Is less than: ________
(a) 
(b) 
(c) 
(d) 

Is at most: __________
(a) 
(b) 
(c) 
(d) 

Is more than: ________
(a) 
(b) 
(c) 
(d) 

The maximum is: ________
(a) 
(b) 
(c) 
(d) 

The minimum is: _________
(a) 
(b) 
(c) 
(d) 
Define a variable and write an inequality to model each situation.
a. He can lift at most 25 pounds with one arm.
___________________
b. You should eat at least 3 servings of vegetables each day. ________________
c. Her goal is to run farther than 5 miles today.
___________________
b. A student has increased his GPA by more than 15 points this quarter.
__________________
2
Using substitution, determine whether each number is a solution of the given
inequality.
1. 3y + 5 < 20
a) 2
b) 0
c) 5
2. 2m  4 ≥ 10
a) -1
b) 8
c) 10
3. 4x + 3 > 9
a) 0
b) -2
c) -4
Graphing Inequalities
For

and

use an open circle.
For

and

use a closed circle.
d

If you put the variable on the left, then the arrow points in the same
direction as the inequality!
If you multiply or divide by a negative, you must switch the inequality sign!
Let’s Practice!
Graphing Inequalities Worksheet
*Do together: #10, #11, #13, #14, #17 & #18*
3
Day 1 Homework
Finish Graphing Inequalities Worksheet & Complete the Problems Below
Write an inequality that represents each verbal expression.
1. a is greater than 4.
a
4
c
3. m is greater or equal to 1.
m
2. c is less than or equal to –2.
1
–2
4. f is less than 2.
f
2
Using substitution, determine whether each number is a solution of the given
inequality. Justify or explain your answer.
5. 2x + 4 < 20
a. 2
b. 10
Define a variable and write an inequality to model each situation.
6. No more than 10 people may use the treadmills at any time in the gym.
Let n =
n
10
7. To train for a marathon, a runner decides that she must run at least 12
miles each day.
Let d =
d
12
4
Day 2 – Solving Multi-Step Inequalities (Day 1)
Determine whether each number is a solution of the given inequality.
1) 2 y  2  10
a. 6
b. 0
c. 11
Is there an easier way to answer the questions above? Explain.
Since there are many (sometimes infinite) solutions, let’s now use our algebra skills
and our knowledge of inequalities to solve each inequality (i.e. find ALL the
solutions)!
Solve and graph the solution on a number line:
1)
2x ≥ -12
Answer: ______________
2) z – 3 < 4
Answer: ______________
5
3) 3x – 7  11
Answer: ______________
4) 4 – 3(x + 5) < 16
Answer: ______________
5) -3(x – 2) + 15 > -9
Answer: ______________
6) 2 y  3  11  2 y
Answer: ______________
Understanding the solution of an inequality.
1. Find the smallest integer that makes the inequality true: 12x  50
Answer: ____________________
2. Find the largest integer that makes the inequality true: 12  2w
Answer: ____________________
3. Find the smallest integer that makes the inequality true when x = 2
10  x  y
Answer: ____________________
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4. A local television station sponsors a food drive. The goal is to donate more
than 1000 canned goods. The station already has collected 400 canned
goods. How many more canned goods does the television station need to
meet its goal? Write and solve an inequality to find the number of canned
goods needed.
Let f be how many more canned goods will come.
5. A family earns at most $2500 a month. The family’s monthly expenses are $2000.
Write and solve inequality to find the possible amounts of money the family could
save each month.
Day 2 Homework
Multi-Step Inequalities Worksheet (#1-12)
7
Day 3 – Solving Multi-Step Inequalities (Day 2)
Warm-Up:
Solve and graph the following multi-step inequalities and write your answers in
inequality notation.
1) 8  4( x  3)  16
Answer: ________________________
2) 7  2 x  23
Answer: ________________________
8
(2 x  4)  6
3)
Answer: ________________________
4) 6x + 4(x -1) > 6(3 + 2x)
Answer: ________________________
5)
Answer: ________________________
6)
Answer: ________________________
9
When you solve an inequality and the variable disappears, you have a special
situation.
Solve each of the inequalities. You will find that you will either have “no solutions”
or “all real solutions”. Write your answer and graph on the number line provided.
Provide justification or explain your answers.
A graph with ‘no solutions’ is blank. A graph of ‘all real solutions’ is everything.
a.) 6w + 5 > 2(3w + 3)
b.) 5r + 15 ≥ 5(r  2)
Answer: _________________
Answer: ___________________
Explain _____________________
Explain _______________________
c)
2(n  3) ≤ 13 + 2n
d) 3(w + 3) < 9  3w
Answer: _________________
Answer: ___________________
Explain _____________________
Explain _____________________
10
Day 3: Homework Worksheet
Solve each inequality. Graph your solution on the number line.
1. 3m + 12 < 24
3. –2 + 2p ≤ –14
2. 4w – 3 ≥ 33
4. 12 > 60 – 6t
Solve each inequality. Write your answers in inequality notation.
5. 4(k + 2) – 3k ≤ 12
6. 3(2c – 2) – 2c > 0
Answer: ________________
Answer: ________________
7. 12(j + 1) + 3j < 57
Answer: ________________
8. 22 ≥ 5(y – 2) – 3y
Answer: ________________
11
Solve each inequality, if possible. If the inequality has no solution, write no
solution. If the solutions are all real numbers, write all real numbers. Graph
your solution.
9. 8w – 5 > 2(4w – 3)
Answer: __________________
10. 4r - 15 ≥ 4(r – 2)
Answer: ________________
11. A grandmother devises an inequality to help her remember the ages of her two
grandchildren. She knows her grandson is two years older than her granddaughter and
that together, they are at least 12 years old. What are the youngest that her grandson
and granddaughter could be?
Let a = the age of the granddaughter
Let a +
= the age of the grandson
12. A family decides to rent a boat for the day. The boat’s rental rate is $500 for the
first two hours and $50 for each additional half hour. Suppose the family budgeted $700
to rent the boat. What is the maximum number of additional half hours for which they
can rent the boat?
Let t = the additional time in half hours.
t + $500
$700
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Day 4 – Interval Notation/Compound Inequalities (Day 1)
Interval Notation
13
x > 2 and x < 10
Includes all numbers between 2 and 10.
Can be written as: 2 < x < 10
x ≤ -3 or x > 2
Includes all numbers less than or
equal to -3 or greater than 2.
The phrase inclusive means that it includes the numbers.
“All numbers between 2 and 10, inclusive”. Can be written as:
2 
x
 10
Express each compound inequality in 1) interval notation and 2) on a number line.
Conjunctions “and” (shade between)
1)
x > -2 and x ≤ 4 (also written as _______________________________ )
Answer: __________________
2) x ≥ -5 and x < 3 (also written as ______________________________ )
Answer: ________________
3) x ≥ -1 and x ≤ 4 (also written as _______________________________ )
Answer: _________________
Disjunctions “or” (shade out …”oars”)
1) x < 4 or x ≥ 7
Answer: ____________________
2) x ≤ -2 or x > 3
Answer: ____________________
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3) x ≤ -1 or x > 4
Answer: ____________________
Solve the following compound inequalities. Then 1) graph the solution 2) write the
solution in interval notation.
1)
0≤x-2≤7
Answer: ________________
2) 3x + 1 < 10 or 3x- 5 ≥ 10
Answer: ________________
3) - 8 < - 4x ≤ 12
Answer: ________________
15
4) - 1 < x + 2 ≤ 4
Answer: ________________
5)
4x – 2 < 10 or 3x + 1 > 22
Answer: ________________
6) 1 ≤ 3x- 2 ≤ 4
Answer: ________________
7) - 2 ≤ - x + 4 < 5
Answer: ________________
16
x
8) 3 + 6 ≤ 3 or 5x - 6 > 24
Answer: ________________
9) - 1 ≤ x + 2 ≤ 6
Answer: ________________
10) - 4 ≤
x
≤3
2
Answer: ________________
Day 4 Homework
Compound Inequalities Practice WS (#1-8)
*Write All Answers in Interval Notation and Graph on the Number Line*
Day 5 – Compound Inequalities (Day 2)
Quiz 1 (Days 1-3)
Classwork/Homework: Compound Inequalities Practice WS (#9-18)
*Write All Answers in Interval Notation and Graph on the Number Line*
17
Day 6: Compound Inequalities - Notation
Classwork/Homework: Using the given information in the chart, fill in all of the blank spaces below.
Inequality
1.
Interval Notation
Graph
x 5
2.
[2, )
3.
4.
1  x  1
5.
(2,3)
6.
7. x  4 or x  0
8.
(, 2)  [3, )
9.
18
Day 7 – Inequality Word Problems (Day 1)
19
1st: Read carefully and underline key words
2nd: Write a let statement
3rd: Determine whether to use , , , or 
4th: Write and solve the inequality
Write the following verbal sentence as an inequality. Then solve for x.
1)
The sum of 13 and x is greater than -26
2)
The difference of x and 18 is less than or equal to 38
3)
Five less than x is greater than 12
4) If 5 times a number is increased by 4, the result is at least 19. Find the least
possible number that satisfies these conditions.
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5) The sum of twice a number and 5 is at most 15. What are the possible values for
the number?
6) The cost of a gallon of orange juice is $3.50. What is the maximum number of
containers you can buy for $15?
7) Three times a number increased by 8 is no more than the number decreased by 4.
Find the largest value for this number.
8) Two-thirds of an integer plus 5 is greater than 12. Find the minimum value for this
integer.
21
Day 7: Homework Worksheet
_____1) 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?
_____2) Which statement is modeled by 2p + 5 < 11?
(1) The sum of 5 and 2 times p is at least 11.
(2) Five added to the product of 2 and p is less than 11.
(3) Two times p plus 5 is at most 11.
(4) The product of 2 and p added to 5 is 11.
_____3) Which is NOT a solution of the inequality 5
(1) 0
(2) 2
 2x   3?
(3) 4
(4) 5
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_____4) Which statement can be modeled by x + 3  12?
(1) Sam has 3 bottles of water. Together, Sam and Dave have at most 12 bottles of
water.
(2) Jennie sold 3 cookbooks. To earn a prize, Jennie must sell at least 12 cookbooks.
(3) Peter has 2 baseball hats. Peter and his brothers have fewer than 12 baseball
hats.
(4) Kathy swam 3 laps in the pool this week. She must swim more than 12 laps.
5) The sum of an integer and 81 is greater than the product of
Find the smallest value for this integer.
6) Four times a number is greater than
 48.
 3 and that integer.
Find the minimum value for this number.
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Day 8 – Inequality Word Problems (Day 2)
Write and solve the following inequality word problems. Show all your work.
1) The length of a rectangle is 5 cm more than its width. The perimeter of the
rectangle is at least 66 cm. Find the minimum integer measures of the length and width.
2) The members of a club agree to buy at least 255
tickets for a theatre party. If they
expect to buy 80 fewer orchestra tickets than balcony tickets, what is the least number
of balcony tickets they will buy?
3) Ashley has a $20 bill and needs to buy three Hallmark Blossom birthday cards at
$2.55 each. With the left over money she would like to buy as many Paw Note thank you
cards as possible. If the thank you cards cost $2.99 each, what is the maximum number
of thank you cards she can purchase?
4) Your new cell phone plan has a monthly access fee of $28.72 with a flat rate of $0.31
per minute. If you want to keep your monthly bill under $40, what is the maximum
number minutes per month you can talk on your new cell phone?
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5) Herman decides to take up golf. His golf club membership will cost $450 for the
season and he will be charged $18 for each round of golf that he plays. Herman has
decided not to spend more than $1000 on golf for the season.
a.) Write an inequality that describes the relationship between the maximum amount
Herman wants to spend and the total golf costs for the season.
b.) Solve the inequality to determine the maximum number of rounds of golf he can
play yet not exceed his $1000 limit.
6) An electronics store sells DVD players and cordless telephones. The store makes a
$75 profit on the sale of each DVD player (d) and a $30 profit on the sale of each
cordless telephone (c). The store wants to make a profit of at least $255 from its sales
of DVD player and cordless telephones.
a.) Write an inequality which represents the situation.
b.) If 2 DVD players were sold, what is the minimum number of cell phones they must
sell to make their desired profit?
7) A prom ticket at Smith High School is $120. Tom is going to save money for the ticket
by walking his neighbor's dog for $15 per week. Tom has already saved $22.
a.) Write an inequality which can be used to find the minimum number of weeks Tom
must walk the dog to earn enough to pay for the prom ticket.
b.) Solve the inequality to determine how many hours Tom must work.
25
8) The low temperatures for the previous two days were 62 degrees and 58 degrees. The
average daily temperature for the three days must be at least 64 degrees.
a) Write an inequality to represents this situation.
b) Solve the inequality to determine what the minimum temperature on the third day
must be.
9) Ike’s age is three years more than twice his younger brother’s age. The sum of their
ages is at most 18.
a) Write an inequality to represents this situation.
b) Solve the inequality to determine the greatest age that Ike’s brother could be.
26
Day 8: Homework Worksheet
Show all work. Make sure you write let statements for each variable.
1. Yellow Cab Taxi charges $1.75 flat rate in addition to $0.65 per mile. Katie has no
more than $10 to spend on a ride.
a)
Write an inequality that represents Katie’s situation.
b) What is the maximum number of miles Katie can travel without exceeding her
limit?
2. Skate Land charges a $50 flat fee for birthday party rental and $5.50 for each
person. Joann has no more than $100 to spend on the birthday party.
a)
Write an inequality that represents Joann’s situation.
b) What is the maximum amount of people Joann can invite to her party without
exceeding her limit?
3. Chris wants to order DVDs over the internet. Each DVD costs $15.99 and shipping for
the entire order is $9.99. Chris has no more than $100 to spend.
a)
Write an inequality that represents Chris’ situation.
b) What is the maximum number of DVDs Chris can order without exceeding his
$100 limit?
27
4. Craig is delivering boxes of paper to each floor of an office building. Each box weighs
64 pounds, and Craig weighs 160 pounds. If the capacity of the elevator is 2000
pounds, what is the maximum number of boxes he can deliver on each elevator trip?
5. Use the given figure. If the area is at least 45 cm2, what is the largest value of x?
28
Day 9 Classwork/Homework – Common Core Practice/Mixed Review
Problems #1-9 are taken from previous Common Core Regents Exams.
1.
2.
3.
4.
29
5.
6.
7.
8.
30
9.
Problems #10-24 - Mixed Review
10. Represent each of the following as an algebraic inequality.
a) x is at most 30
___________________
b) the sum of 5x and 2x is at least 14
___________________
c) the product of x and y is less than or equal to 4
___________________
d) 5 less than a number y is under 20
___________________
11. Find the smallest integer that makes the inequality true.
10  5y
12. Find the largest integer that makes the inequality true when y = 2.
8  a  ay  y  5
31
Solve each inequality and graph its solution on the number line.
32