Algebra 1A UNIT 3: INEQUALITIES
LESSON 3-1: Inequalities and Their Graphs
LESSON 3-2: Solving Inequalities Using Adding or Subtracting
LESSON 3-3: Solving Inequalities Using Multiplying or Dividing
LESSON 3-4: Solving Multi-Step Inequalities
LESSON 3-1: Inequalities and Their Graphs
OBJECTIVE: To write, graph, and identify solutions of inequalities.
An __________________ mathematical sentence that contains inequality symbol to compare the
value of two expressions.
You can use a number line to visually represent that values that satisfy an inequality.
Symbol
>
Algebraic Meaning
Graph
<
≥
≤
< and > use ______ CIRCLES and the number is _____________ as an answer.
≤ and ≥ use _________ CIRCLES and the number is ____________ as an answer.
A SOLUTION OF AN INEQUALITY is any value that makes the inequality ______.
Examples: 6, 8, and 15 are solutions of x ≥ 6 because 6 ≥ 6, 8 ≥ 6, and 15 ≥ 6
Non-Examples: 3 and -1 are not solutions of x ≥ 6 because 3 ≥ 6 and -1 ≥ 6
Examples 1a – 1f: Write an inequality to represent each variable expression.
a.) All real numbers x less than or equal to -7.
b.) 6 less than a number k is greater than 13.
c.) All real numbers x greater than 5.
d.) 4 greater than a number y is less than 10.
e.) The quotient of t and 7 is less than -3.
f.) 3 times a number n greater than or equal to -9.
Examples 2a – 2d: Decide whether the number below is a solution of 2x + 1 > -3.
a.) -3
b.) -1
c.) -2
d.) 1
Examples 3a – 3d: Decide whether the number below is a solution of 13 – 7y ≤ 6.
a.) -1
b.) 0
c.) 1
d.) 3
Examples 4a – 4d: Decide whether the number below is a solution of 3x - 2 < -1.
a.) 4
b.) -3
c.) 0
d.) -4
Examples 5a – 5d: Graph the solution of each inequality.
a.) n ≤ -3
b.) m > -5
c.) 2 < r
d.) 3 ≥ n
Examples 6a – 6d: Write an inequality for each graph.
a.)
b.)
c.)
d.)
Examples 7a – 7d: Write an inequality to describe each situation. Be sure to define a variable.
a.) You must be at least 18 years to vote.
b.) You must be at least 4 feet to ride this roller coaster.
c.) Speed limit 35 mph.
d.) Trail Rides starting at $19.99
LESSON 3-2: Solving Inequalities Using Adding or Subtracting
OBJECTIVE: To use addition or subtraction to solve inequalities.
BELL RINGER: Write and graph an inequality for “To qualify for a race, your time can be at most 62
seconds.” Be sure to define your variable,
NOTES
EQUIVALENT INEQUALITIES are inequalities that have the SAME solutions.
The process of solving inequalities is similar to solving equations: Use inverse operations.
PROPERTIES OF INEQUALITIES:
ADDITION PROPERTY OF INEQUALITIES:
You can ______ the _______ value to ________ side of an inequality.
Example:
Algebra:
Example:
Algebra:
SUBTRACTION PROPERTY OF INEQUALITIES:
You can ______________ the ________ value to ________ side of an inequality.
Example:
Algebra:
Example:
Algebra:
The PROPERTIES are the SAME for ≥ and ≤.
Examples 1a – 1f: Solve each inequality. Graph and check your solution.
a.) n – 10 > -14
b.) y – (-7) ≥ 12
c.) 10 ≥ x – 3
d.) d + 9 ≤ 8
e.) 7 ≤ x + 3
f.) q – 2 > -6
Example 2: A school bus can safely carry as many as 76 students. If 19 students are already on the
bus, how many more can board the bus?
Example 3: The hard drive on your computer has a capacity of 120 GB. You have used 85 GB. You
want to save some home videos to your hard drive. What are the possible sizes of the home video
collection you can save?
Example 4: To get an A, you need more than 200 points on a two-part test. You scored 109 on the
first part. How many more points do you need?
Example 5: A club has a goal to sell at least 25 plants for a fundraiser. Club members sell 8 plants
on Wednesday and 9 plants on Thursday. What are the possible numbers of plants the club can sell
on Friday to meet their goal?
Example 6: Find the mistake. Then solve the inequality to find the correct answer.
x – (-2) < 3
x+2<3
-2 -2
x<1
0
LESSON 3-3: Solving Inequalities Using Multiplying or Dividing
OBJECTIVE: To use multiplication and division to solve inequalities.
BELL RINGER: Solve the inequality 8 ≥ d + 5. Graph and check your solution.
NOTES
PROPERTIES OF INEQUALITIES:
DIVISION PROPERTY OF INEQUALITIES:
If you _________ each side of an inequality by the same ____________ number, the direction of the
inequality ________ remains _____________.
Example:
Algebra:
Example:
Algebra:
If you __________ each side of an inequality by the same _____________ number, the direction of
the inequality ___________ is _____________.
Example:
Algebra:
Example:
Algebra:
MULTIPLICATION PROPERTY OF INEQUALITIES:
If you ______________ each side of an inequality by the same _____________ number, the direction
of the inequality ___________ remains ____________.
Example:
Algebra:
Example:
Algebra:
If you _____________ each side of an inequality by the same _____________ number, the direction
of the inequality ____________ is _____________.
Example:
Algebra:
Example:
Algebra:
The PROPERTIES are the SAME for ≥ and ≤.
Why it works: Take 3 > 1 and multiply it by -2 and see what happens.
Examples 1a – 1h: Solve each inequality. Graph and check your solution.
a.) 6a ≤ -18
b.) -8m ≥ -24
c.) -4p < 36
e.) 
g.)
y
 2
8
s
3
10
d.) -21 < 7p
f.) 2  
2
w
3
3
4
h.)  w  3
Example 2: A woodworker makes a profit of $30 on each picture frame sold. How many frames must
he sell to make a profit of at least $500?
Example 3: You walk dogs in your neighborhood after school. You earn $4.50 per dog. How many
dogs do you need to walk to earn at least $75.
Example 4: Your class is taking a trip to a museum that is 190 miles away. The bus can travel at 55
miles per hour. At least how many hours should your class plan for the trip to the museum?
Example 5: A cheerleading squad earns $5.50 per car washed. How many cars does the squad
need to wash to earn at least $77?
Example 6: A student club plans to buy food for a soup kitchen. A case of vegetables cost $10.68.
The club can spend at most $50 for this project. What are the possible number of cases the club can
buy?
Example 7: Find the mistake. Then solve the inequality to find the correct answer.
-2x > 10
-2x > 10
-2 -2
x > -5
0
Example 8:
LESSON 4: Solving Multi-Step Inequalities
OBJECTIVE: To solve multi-step inequalities.
BELL RINGER: Solve the following inequalities. Graph and check your solution.
1
a.) 32 ≤ 8t
b.)  x  7
3
NOTES
How are to solve multi-step inequalities with variables on both sides?
1.) Simplify each side of the inequality (distribute, combine like terms, etc.)
2.) Move the variables to one side of the inequality and combine like terms.
3.) Move the constants to the other side of the inequality and combine like terms.
4.) Solve and graph the resulting inequality.
REMEMBER when you _________________ or ______________
by a ________________ ______________,
the sign is__________________!!!!
Inequalities can have 3 types of solutions:
*ONE solution
*NO solution
*INFINITE number of solutions
(ALL REAL NUMBERS)
Example:
Example:
Example:
Examples 1a – 1p: Solve each inequality.
a
a.)  3  2
4
b.)  5 
c
 1
3
c.) -3.5x – 6 > -1.8
e.)
1
1
1
 a
5
3
2
d.) -6.1 + 3.4d < 7.5
f.) -4 < 5 – 3a
g.) 3(t + 1) – 4t ≥ -5
h.) 15 ≤ 5 – 2(4m + 7)
i.) 6n – 1 > 3n + 8
j.) 3b + 12 > 27 – 2b
k.) 10 – 8a ≥ 2(5 – 4a)
l.) 6m – 5 > 7m + 7 - m
m.) 4p + 3 > 9p + 6 – 5p
n.) 8 + 6x ≥ 7x + 2 – x
o.) 9 + 5n ≤ 5n – 1
p.) 14 – 6n ≤ 2(7 – 3n)
NOTE:
If the variables cancel out, then look at the arithmetic
statement and determine whether it is true or false.
TRUE → All Real #s
FALSE → No Solution
Example 2: A music club charges $.75 per song download plus a membership fee of $5.70. Diego
can spend at most $15. What is the greatest number of songs that Diego can download?
Example 3: A phone plan charges $.20 per text message plus a monthly fee of $42.50. Lin can
spend at most $50. Write an inequality for the number of text messages Lin can spend. Describe the
solution.
Example 4: you want to fence in a rose bed that is adjacent to the deck. The length of your deck is
14 ft. You have at most 44 ft. of fence. What are the possible widths of the rose bed?
Example 5: You want to make a rectangular banner that is 18 ft long. You have no more than 48 ft.
of trim for the banner. What are the possible widths of the banner?
Example 6: In a community garden, you want to fence in a vegetable garden that is adjacent to your
friend’s garden. You have at most 42 ft. of fence. What are the possible lengths of your garden?
Example 7: The Fun Zone charges $5 for admission plus $.25 per token for arcade games. Eric has
saved at least $10 from doing chores. How many tokens can Eric buy?
Example 8: You want to buy a new tablet that costs at least $450. You have already saved $130.
You want to save the same amount each month for the next 4 months to have enough money for the
tablet. What is the least amount of money you need to save each month?
Example 9: You want to buy some T-shirts and a pair of shorts. At one store, T-shirts cost $10 each
and shorts cost $20 each. You do not want to spend more than $85. How many T-shirts can you
buy?
Example 10: Find the mistake. Then solve the inequality to find the correct answer.
5p + 15 > -4p + 12
-4p
-4p
1p + 15 > 12
-15 -15
1p > -3
p > -3
0