M3: Chapter 3 Notes
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Academic
Pre-Algebra
Chapter 3 Notes
Multi-Step
Equations and
Inequalities
Name_________________ Pd._____
M3: Chapter 3 Notes
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Section 3.1: Solving Two-Step Equations
Learning Goal: We will solve two-step equations.
**When solving two-step equations, we will use ______________________ in
____________________________.
Example 1: Using Subtraction and Division to Solve
Solve 8 x  7  47 . Check your solution.
ON YOUR OWN:
Solve the equation. Check your solution.
1. 4 x  1  5
2. 3n  8  2
4.
2  6h  20
5.
3x  7  5
3.
1  2r  9
M3: Chapter 3 Notes
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Example 2: Using Addition and Multiplication to Solve
Solve
y
 2  1 . Check your solution.
4
ON YOUR OWN:
b
8 1
1.
4
4.
12 
f
8
2
c
26
2.
6
5.
3. 2 
d
1
5
x
 3 1
2
Example 3: Solving an Equation with Negative Coefficients
Solve 9  6a  45 . Check your solution.
M3: Chapter 3 Notes
ON YOUR OWN:
Solve the equation. Check your solution.
1. 12  4s  12
2. 6  2m  8
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3.
2  5n
Example 4: Writing and Solving a Two-Step Equation
You are buying a drum set that costs $495. The music store lets you
make a down payment. You can pay the remaining cost in three equal
monthly payments with no interest charged. You make a down payment
of $150. How much is each monthly payment?
ON YOUR OWN:
Alex makes a down payment of $1100 on a motorcycle costing $2300.
If no interest is charged, how many monthly payments of $200 must he
make until he has finished paying for the motorcycle?
M3: Chapter 3 Notes
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Section 3.2: Solving Equations Having Like Terms and
Parentheses
Learning Goal: We will solve equations using the distributive property and by
combining like terms.
Example 1: Solving an Equation by Combining Like Terms
Solve 13t  7  10t  2 . Check your solution.
ON YOUR OWN:
Solve the equation. Check your solution.
2. 2d
1. 22  4 y  14  0
 24  3d  84
Example 2: Writing and Solving an Equation
One basketball team defeated another by 13 points. The total number of points
scored by both teams was 171. Write an equation to represent this situation.
Solve the equation to find the number of points scored by the winning team (p).
M3: Chapter 3 Notes
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Bill and Jasmine together have 94 glass marbles. Bill has 4 more than twice as
many marbles as Jasmine. Jasmine has m marbles. Write an equation to represent
the situation. Solve the equation to find how many glass marbles each has.
ON YOUR OWN:
A bookstore spent $241 to send a group of students to a reading competition.
Each student who won was given a $5 gift certificate and a personalized bookmark
that cost $2. Included in the $241 was $45 for the salary of the staff member
who accompanied the students to the competition. How many students won prizes?
Example 3: Solving Equations Using the Distributive Property
Solve the equation.
1.  21  7(3  x)
2.  3(8  4 x)  12
3.
26  2( x  6)
4.
 2(17  3k )  22
M3: Chapter 3 Notes
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STEPS FOR SOLVING A MULTI-STEP EQUATION:
STEP 1:
STEP 2:
STEP 3:
STEP 4:
Example 4: Combining Like Terms After Distributing
Solve the equation.
1. 38  3(4 y  2)  y
2. 5 x  2( x  1)  8
ON YOUR OWN:
Solve the equation. Check your solution.
1. m  4(2m  3)  3
2. 13  2 y  3( y  4)
M3: Chapter 3 Notes
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Section 3.3: Solving Equations with Variables on Both
Sides
Learning Goal: We will solve equations with variables on both sides.
**You can solve equations with variables on both sides by getting the
__________________________ on one side of the equation and the
__________________________ on the other side.
Example 1: Solving an Equation with the Variable on Both Sides
Solve 7n  5  10n  13 .
Solve
5k  8  7k  18 .
M3: Chapter 3 Notes
ON YOUR OWN:
Solve the equation. Check your solution.
1. 5n  2  3n  6
2. 8 y  4  11 y  17
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3.
m  1  9m  15
Example 2: Writing and Solving an Equation
A bus tour of celebrities’ homes costs $300 for the bus and tour guide
plus $8 per person for lunch. How many people does a tour group need
to have in it so that the cost per person is $20?
ON YOUR OWN:
The Spanish club is arranging a trip to a Mexican restaurant in a
nearby city. Those who go must share the $60 cost of using a school
bus for the trip. The restaurant’s buffet costs $5 per person. How
many students must sign up for this trip in order to limit the cost to
$10 per student?
M3: Chapter 3 Notes
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**All of the equations we have dealt with until now have had one
solution. However, this will not always be the case when solving
equations.***
Number of Solutions:
1.
2.
3.
Example 3: An Equation with No Solution
Solve 4(2  3x)  12 x .
ON YOUR OWN:
Solve 5(2 x  1)  10 x .
Example 4: Solving an Equation with All Numbers as Solutions
Solve 6 x  2  2(3x  1) .
M3: Chapter 3 Notes
ON YOUR OWN:
Solve 7(2 x  3)  21  14 x .
Example 5: Solving an Equation to Find a Perimeter
Find the perimeter of the square.
Step 1:
Step 2:
Step 3:
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M3: Chapter 3 Notes
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Section 3.4: Solving Inequalities Using Addition or
Subtraction
Learning Goal: We will solve inequalities using addition or subtraction.
Vocabulary:
 Inequality – a mathematical statement formed by placing an
inequality symbol between two expressions
 Solution of an inequality – the set of all numbers that produce
true statements when substituted for the variable in the
inequality
**When you graph an inequality of the form x  a or x  a , use an
___________ circle at a.
**When you graph an inequality of the form
x  a or x  a , use a
___________ circle at a.
Inequality
Words
x3
All numbers less than 3
y2
z4
n2
Graph
M3: Chapter 3 Notes
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Example 1: Writing and Graphing an Inequality
A cyber café charges users a minimum fee of $2 for internet access.
Write an inequality to represent the access fee f. Then graph the
inequality.
ON YOUR OWN:
The freezing point of water is 0 degrees Celsius. At temperatures at
or below the freezing point, water is a solid (ice). Write an inequality
that gives the temperatures at which water is a solid. Then graph the
inequality.
 Equivalent inequalities – inequalities that have the same solution
M3: Chapter 3 Notes
Example 2: Solving an Inequality Using Subtraction
Solve k  1  4 . Graph and check your solution.
Example 3: Solving an Inequality Using Addition
Solve  1  y  7 . Graph and check your solution.
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M3: Chapter 3 Notes
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ON YOUR OWN:
Solve the inequality. Graph and check your solution.
1. n  7  3
2.
6  x 9
Example 4: Writing and Solving an Inequality
On the first two tests in math class, Ryan had scores of 89 and 95 points. The
third math test is tomorrow, and Ryan’s goal is to have a total score of 279 or
higher on the three tests in order to have an A average for this quarter. What
possible scores s can he have on the test tomorrow to attain his goal?
M3: Chapter 3 Notes
ON YOUR OWN:
Extra Practice:
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M3: Chapter 3 Notes
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Section 3.5: Solving Inequalities Using Multiplication or
Division
Learning Goal: We will solve inequalities using multiplication or division.
**When each side of the inequality is multiplied by a positive number,
the inequality remains ___________________________________.
**When each side of the inequality is multiplied by a negative number
the inequality sign ___________________________________.
M3: Chapter 3 Notes
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Example 1: Solving an Inequality Using Multiplication
k
 5 .
a. Solve
8
m
 3.
b. Solve
3
**The rules for solving an inequality using division are the same as the
rules for solving an inequality using multiplication.
Example 2: Solving an Inequality using Division
b. Solve  10t
a. Solve  18 y  72 .
 34 .
M3: Chapter 3 Notes
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ON YOUR OWN:
Example 3: Writing and Solving an Inequality
On average, a customer service representative helps at most 160
customers during an 8-hour workday. Write and solve an inequality to
find the average number of customers c she can help each hour during
one of her workdays.
M3: Chapter 3 Notes
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Section 3.6: Solving Multi-Step Inequalities
Learning Goal: We will solve multi-step inequalities.
Example 1: Writing and Solving a Multi-Step Inequality
Your school’s soccer team is trying to break the school record for goals
scored in one season. Your team has already scored 88 goals this
season. The record is 138 goals. With 10 games remaining on the
schedule, how many goals, on average, does your team need to score per
game to break the record?
ON YOUR OWN:
A city’s record rainfall for the month of October is 16.8 inches. So far
in October this year, 11.2 inches of rain have fallen. Find the average
number of inches of rain that must fall each day to break the record if
there are 14 days left in the month.
M3: Chapter 3 Notes
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Example 2: Solving a Multi-Step Inequality
a. Solve  6 
g
2.
5
ON YOUR OWN:
Solve the inequality.
1. 7  5 x  3
2.
b. Solve
5 y  2  y  34
x
 6  5 .
4
3. 10  6 
y
5
Example 3: Combining Like Terms in a Multi-Step Inequality
A dance group charges $15 for membership. Members pay $8 for
admission to monthly dances, and nonmembers pay $12. How many
dances does Devon need to attend for the cost of membership to be
less than paying as a nonmember?
M3: Chapter 3 Notes
ON YOUR OWN:
Extra Practice:
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