Unit 2b – Algebraic Thinking –
Algebraic Equations and Inequalities
Class Notes
Date
Equations and Their Solutions
Learning Target: I can determine if value is a solution to an equation.
Key Terms


Equation: A mathematical sentence that shows that two expressions are equal.
Solution of an equation: A value or values that make an equation true.
Steps
1. Substitute in for the variable.
2. Evaluate
3. If both sides are equal then yes, this is a solution; if both sides are not equal then no, this is not
a solution.
Examples
Try This
1)
c + 23 = 48 for c = 35
2) 75 ÷ y = 5 for y = 15
2) 96 = 130-d for d = 34
4) 105 = 7p for p = 14
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Page 1
Date
Solving Addition Equations
Learning Target: I can solve one-step addition equations.
Key Terms


Isolate the Variable: To get the variable by itself in an equation.
Inverse Operations: Operations that undo each other. For example, addition and subtraction,
multiplication and division.
Steps
1. Circle the variable. If you choose, divide the equation into two parts.
2. Isolate the variable by using inverse operations. Remember that whatever operation you do to
one side of the equation; you have to do the same operation to the other.
3. Bring down the equals sign and simplify on both sides.
4. Check your work.
Examples
Word Problem
Kaitlin is 2 inches taller than Reba. Kaitlin is 54 inches tall. How tall is Reba?
Reba’s
Height
x
2 inches
+
-
2
2
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Kaitlin’s
Height
=
54
=
-2
x =
52
Reba is 52 inches in height
Page 2
Try These
1) 𝑥 + 54 = 90
2) 𝑛 + 27 = 46
3) 49 = 12 + 𝑦
4) 22 + 𝑡 = 91
5) 𝑦 + 13.82 = 24
6) 11.4 = ℎ + 5.9
1
2
7) 𝑘 + 5 10 = 7 5
3
8) 𝑥 + 4 10 = 7
9) Lou, Michael and Georgette live on Mulberry Street. Lou lives 10 blocks from Georgette.
Georgette lives 4 blocks from Michael. How many blocks does Michael live from Lou?
10) You build 7 model airplanes during the summer. At the end of the summer, you have 25 model
airplanes, how many model airplanes did you have before the summer?
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Page 3
Date
Solving Subtraction Equations
Learning Target: I can solve one-step subtraction equations.
Key Terms


Isolate the Variable: To get the variable by itself in an equation.
Inverse Operations: Operations that undo each other. For example, addition and subtraction,
multiplication and division.
Steps
1. Circle the variable. If you choose, divide the equation into two parts.
2. Isolate the variable by using inverse operations. Remember that whatever operation you do to
one side of the equation; you have to do the same operation to the other.
3. Bring down the equals sign and simplify on both sides.
4. Check your work.
Examples
Word Problem
Bruce has 25 CD’s remaining after giving 14 to John, 17 to Mary and 25 to Sue. How many CD’s did
Bruce begin with?
Bruce’s
CD’s
(simplify)
c
c
CD’s given
away
+
(14 + 17 + 25)
56
56
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CD’s
remaining
=
=
=
25
25
+ 56
c =
81
Bruce had 81 CD’s
Page 4
Try These:
1)
𝑝– 8 = 9
2) 𝑛 – 24 = 9
3) 3 = 𝑥 – 16
4) 39 = 𝑑 – 2
5) 𝑛 − 10.5 = 11.7
6) 8.77 = 𝑝 − 1.23
7)
6
10
9
= ℎ − 2 10
1
8) 3 5 + 𝑦 = 14
9) Susan is taller than James. The difference in their height is 12 inches. James is 62 inches tall.
How tall is Susan?
10) You buy several posters. The total cost is $18.95. You have $7.05 left after you pay. How much
money did you have before this purchase?
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Page 5
Date
Solving Multiplication Equations
Learning Target: I can solve one-step multiplication equations.
Key Terms


Isolate the Variable: To get the variable by itself in an equation.
Inverse Operations: Operations that undo each other. For example, addition and subtraction,
multiplication and division.
Steps
1. Circle the variable. If you choose, divide the equation into two parts.
2. Isolate the variable by using inverse operations. Remember that whatever operation you do to
one side of the equation; you have to do the same operation to the other.
3. Bring down the equals sign and simplify on both sides.
4. Check your work.
Examples
Word Problem
The area of a rectangular basketball court is 8250 m². If its width is 75 m, what is its length?
(Draw a picture to help)
length = y (unknown)
Area = length (width)
8250 = y (75)
width =
.
75
75
Total Area = 8250 m²
75 m
110 = y
The width of the gym is 110 m.
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Page 6
Try This
1) 7𝑥 = 21
2) 90 = 10𝑎
3) 56 = 7𝑏
4) 4𝑦 = 0
5)
2.5𝑔 = 17.5
3
7) 3𝑥 = 12 5
6) 5.6𝑘 = 19.152
8)
4
𝑥
10
=1
9) The area of a rectangular deck is 675 square feet. The deck’s width is 15 feet. What is the
length?
10) An egg carton holds 12 eggs. One day a farmer gathers 8616 eggs. How many cartons are
needed for all the eggs?
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Page 7
Date
Solving Division Equations
Learning Target: I can solve one-step division equations.
Key Terms


Isolate the Variable: To get the variable by itself in an equation.
Inverse Operations: Operations that undo each other. For example, addition and subtraction,
multiplication and division.
Steps
1. Circle the variable. If you choose, divide the equation into two parts.
2. Isolate the variable by using inverse operations. Remember that whatever operation you do to
one side of the equation; you have to do the same operation to the other.
3. Bring down the equals sign and simplify on both sides.
4. Check your work.
Examples
Word Problem
The seats in the theater are divided into 6 sections. There are 35 seats in each section. How many seats
are there in the entire theater?
Total
sections
Total seats
s
÷
×
6
6
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Seats per
section
=
35
=
×6
x =
210
The theater has 210 seats.
Page 8
Try This
1)
3)
5)
7)
𝑐
12
=8
30 =
𝑦
1.6
1
3
𝑐
6
= 0.256
𝑥=
5
6
2)
4)
6)
8)
49 =
𝑐
24
ℎ
2.4
1
𝑦
4
𝑐
3
= 18
= 15
1
= 32
9. There are 16 ounces in a pound. A box of nails weighs 4 pounds. How many ounces does the
box weigh?
10. The area of Danielle’s garden is one-twelfth the area of her entire yard. The area of the garden
is 10 square feet. Find the area of her yard.
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Page 9
Date
Graphing Simple Inequalities
Learning Target: I can graph simple inequalities on a number line.
Key Terms



Inequality: A mathematical sentence that shows the relationship between quantities that are
not equivalent.
Algebraic inequality: An inequality that contains a variable.
Solution set: The set of values that make a statement true.
Important Information
Examples
Try This
1) w ≤ 0
2) m > -3
3) -1 ≥ x
4) 4 < t
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Page 10
Date
Solve and Graph One-step Inequalities
Learning Target: I can solve and graph one-step inequalities.
Important Information
 You solve inequalities the same way you solve equations.
o However, this will slightly change with the introduction of integer operations next year.
 Remember that…
o Less than (<) and greater than (>) are represented with an open dot when you graph
them.
o Less than or equal to (≤) and greater than or equal to (≥) are represented with a closed
dot when you graph them.
Examples
Try This
Solve and graph each inequality.
1) 3t ≤ 27
2) y – 5 ≥ 0
3) x + 4 < 10
4)
5) 16 < f - 11
6) 8 > 2k
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Page 11
Date
Writing Inequalities to Solve Real-World Problems
Learning Target: I can write inequalities to solve real-world problems.
Key Terms

Inequality: A mathematical sentence that shows the relationship between quantities that are
not equivalent.
Important Information
Examples
Try These
1) No more than 18 people allowed in the gallery at one time.
2) There are fewer than 8 fish in the aquarium.
3) The water level is above 45 inches.
4) The temperature is below 40°F.
5) There are at least 24 pictures on the roll of film.
6) We can take at most 32 students on the bus.
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Page 12
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Page 13