1
Unit 3
Linear Equations
& Inequalities
Created by: M. Signore & G. Garcia
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Lesson #13: Solving One Step Equations
Do Now:
1. Which sentence illustrates the distributive property?
a) xy = yx b) x(yz) = (xy)z
c) x(y + z) = xy + xz d) 1(xy) = xy
2. What is the additive inverse of -4a?
a)
4
a
b) 4a c) -
4
a
d) -
4a
1
3] What is the reciprocal of 5?
4] What is the reciprocal of
7
2
?
5] In the step-by-step simplification of the expression below, which
property is not used?
a) Commutative b) Distributive c) Associative d) Identity
3(1 + x)
3(x + 1)
3
x
+ 3
1
3
Simplify the following shown below:
6]
x + x =
7]
x
x =
8] (3xy
3
z
2
)
3
9] (x
–
4)
2
Solving Equations
An equation is a mathematical statement that two expressions have
the same value, which are separated by an equal sign, “=”.
“Algebra is the art of reducing and solving equations.” - Al-Khwarizmi
Algebra has been used for over 4500 years dating back to
ancient Babylon, where assignments were written on clay tablets using
the ends of little sticks to make wedge-shaped marks. About 1000
years later, Egyptian students wrote their assignments on papyrus, a
parchment-like material that was easier to
write on than a clay tablet.
The Greek mathematician, Diophantus, introduced his style of
writing equations.
4
Steps for Solving ALL Equations:
1
st
Distribute to get rid of the parentheses, if necessary.
2
nd
Combine like terms on the same side of the “=”.
3
rd
Add & subtract to get all constants to one side of the “=” and all
variables to the other side of the “=”.
4
th
Multiply & Divide to get the variable by itself.
5
th
Check using substitution.
For Solving ALL EQUATIONS, you need to add, subtract, multiply,
and/or divide from BOTH sides of the equation in order to
ISOLATE the given variable.
One-Step Equations:
Adding/Subtracting on Both Sides:
x + 14 = 21
Check:
-
13 =
-
4 + h
Check:
To Check, substitute your SOLUTION into the ORIGINAL
equation!
Simply, “undo”
what has been
done to the
5
QUICK CHECK: Solve the following equations
k + 11 =
-
21
Check:
h
–
26 =
-
29
Check:
–
12 + z =
-
36
Check:
–
23 =
-
19 + n
Check:
Multiplying/Dividing on Both Sides:
5y = 30
Check:
4
8
x
Check:
To Check, substitute your SOLUTION into the ORIGINAL
equation!
6
QUICK CHECK: Solve the following equations
–
4r =
-
28
Check:
11 =
5
x
Check:
14 = -
3
7
a
Check:
5x =
-
45
Check:
Independent Practice: Solve and check the following equations
x + 2 = 10
Check:
3x =
-
15
Check:
3
x
= -6
Check:
-
7
=
-
16
-
k
Check:
2
16
3
X
Check:
-
8 = x
–
7
Check:
7
Homework #13: Solving One Step Equations
Directions: Solve and check each of the following equations. Show all of your work!
Solve Check
1)
y – 4 = 11 or 11 = y – 4
2)
n + 40 = 25 or 40 + n = 25
3)
– 3q = 10
8
Solve Check
4)
– b = 35 or _____= 35
5)
5
d
= 11
6)
4
3
k = 12
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Lesson #14: Solving Two Step Equations
DO NOW:
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Two-Step Equations:
Your turn! Be sure to show all of your steps
2x
–
10 = 18
10 + 7X = 45
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Independent Practice:
4x + 5 = -27
Check it!
5
x
- 20 = 35
-
4k + 15 =
-
1
5
3
x – 10 = 20
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HW #15: Solving Two Step Equations
Directions: Solve for x and check. Show all work!
Solve Check
1)
4x + 5 = 29
2)
7 – 5x = 22
3)
30 – x = 40
4)
812
6
x
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Solving Two Step Equations Regents Test
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15
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Lesson #16: Solving Multi-Step Equations
DO NOW:
Solve and check the following equations:
1)
9 11
5
y
2)
61 7 26y
Solving Multi-step equations: A multi-step equation requires you
to combine like terms first.
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Guided Example:
A)
n + 4n
-
11 = 19
B)
9
-
y + 6y =
-
6
C)
60
-
12b + 12 = 0
D) 8x – 4x +15 = -5
E) 4a + 2 – 12a = 18
You Try
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F) -10c + 5 -8c = 59 G) -10 – 14y +21 = 53
H) 12 + 6p +3 = 63 I) 3 – 7a +6a -4 = 8
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Homework #15: Solving Multi-Step Equations
1) 7a +3a +2 –a = 20 2) 5m – 18 -6m = 4
3) -5y -3y +4 = -4 4) 8x – 3x -10 = 20
5) 9d -2d +4 = 32 6) 9 = 7z -13z -21
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Lesson #16: Solving Equations with the Distributive
Property
SOLVE CHECK
1)
56138
x
56138
x
56)1(8)3(8
x
56824
x
8
8
4824
x
24
24
2
x
56138
x Rewrite
561)2(38
Replace
56168
Recalculate
5678
5656
2)
80138
x
3)
)10(230 y
“Replace” variable
with your answer
*
Distribute
* Keep “+” sign
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SOLVE
CHECK
4)
27479
x
5)
3093
y
6)
25)68(
2
1
x
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Homework #16: Solving Equations with the
Distributive Property
Solve and check each equation!
SOLVE CHECK
1)
108724
x
2)
3353
x
3)
1826
x
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Lesson #17: Solving Equations with Variables on
Both Sides
SOLVE CHECK
1)
xx 2449
xx 2449
x2 x2
4411 x
11
11
3) Now you only have your
variable on one side of the
equation
4
x
xx 2449
Rewrite original
)4(244)4(9
Replace solution
84436
Recalculate
3636
2)
cc
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Steps
:
1) Variables on both sides
2) Perform inverse operation
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3)
xx
729
4)
503107
rr
5)
dddd
539126
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Homework #17:Solving Equations with Variables on
Both Sides
SOLVE CHECK
1) 7n - 3 = n + 27
2)
141230
yy
3)
35547
xxx
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Solving Equations with Variables on Both Sides
Regents Test
1.
15
−
24
−
4
=
−
79
2.
102
=
69
−
7
+
3
3.
3
2
−
5
−
4
=
33
4.
3
4
−
5
+
2
11
−
2
=
43
5.
9
3
+
6
−
6
7
−
3
=
12
6.
7
4
−
5
−
4
6
+
5
=
−
91
7.
−
12
=
−
6
8.
3
−
25
=
11
−
5
+
2
27
9.
10.
11.
12.
13.
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Lesson #18: Solving Literal Equations
Do Now:
1]
Evaluate x
2
–
4y
2
,
when, x =
-
2
and y = -8.
2] Simplify: (x
–
2)
2
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A literal equation is an equation that contains two or more variables.
You can use inverse operations to solve for one variable in terms of
others (aka transforming formulas).
y = 2x + 3
v
m
D
Formula or Literal equation?
___________
What is it being solved for?
________
Example:
Given 2y + 5x = 16, solve for y in terms of x.
What
is it
being
solved for?
Is y
–
2x = 3
equivalent?
In Science,
Isolate
_____
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Example: Given ax + b= c, express a in terms of b and c
Distance Formula: Distance = rate • time
Example: Given D = r•t, solve for the following shown below,
Solve for r:
Solve for t:
Example: Given A =
1
2
bh, solve for h.
Steps:
A =
1
2
bh
Divide by b
Isolate
____
Formula
for Area
of a
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Temperature:
The formula, F=
9
5
C + 32 gives the temperature F in degrees
Fahrenheit in terms of a given temperature C in degrees Celsius. Solve
for C in terms of F.
January 2011 Regents Questions:
If
tk
n
ey
, what is y in terms of e, n, k, and t?
You Try:
1)
pforsolveprtpA
2)
rforsolvehrV
2
3
1
3)
wforsolvewlP 22
4)
xforsolvebf
d
cx
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HW #18: Solving Literal Equations
Solve the following for the indicated variable shown below:
1] Solve for x:
2] Solve for p:
3ax + b = c
2m + 2p =
16
3] Solve for p:
4] Solve for y:
A =
2
1
p + rt
-
4x + 2y = 12
5] The formula for potential energy is P = mgh, where P is potential
energy, m is mass, g is gravity, and h is height. Express g in terms of P,
m, and h
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Literal Equations Regents Test
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35
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Lesson #19: Translating Algebraic Expressions
Do Now: Place the following words in the appropriate boxes
Word Bank: more than decreased by take away increased by
Diminished total plus minus exceeds
Greater than sum difference less than
Added to
Addition
Subtraction
3 more than
x
the sum of 10 and a number c
a number n increased by 4.5
a number
t
decreased by 4
the difference between 10
and a number y
6 less than a number z
Examples
Highlight the words that indicate what operation to use and UNDERLINE the numbers.
1. The sum of a number and 10 ___________ 3. The difference of a number and 2 _______
2. 9 less than an number _______________ 4. A number increased by 7 ___________
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Word Bank: quotient times product divided by multiplied by
Twice double of divide triple half
Multiplicati
on
Division
the product of 3 and a
number t
twice the number x
4.2 times a number e
the quotient of 25 and a
number b
the number y divided by 2
Examples
Highlight the words that indicate what operation to use and UNDERLINE the numbers
1. the quotient of a number and 5 __________ 3. The product of a number and -7 ________
2. twice a number plus 8 __________ 4. Half of a number minus 1 __________
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You Try: Write an algebraic expression for each verbal expression
1. eight less than a number 2. a number increased by seven
3. the quotient of m and n 4. a number squared
5. nine times a number 6. a number decreased by three
7. seven more than the cube of a number 8. one-half the product of x and y
9. the product of twice a and b *10. twice the product of a and b
11. two less than five times a number
12. twice a number increased by 3 times that number
*13. the sum of 3 times a and b *14. three times the sum of a and b
Write a verbal expression for each algebraic expression
15. 3x – 4 16. x + 7
17.
x
y
18. ½ (x + y)
19. x – 6 20. 8y
2
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Homework #19: Translating Algebraic Expression
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Lesson #20: Coin Word Problems
Do Now:
41
42
43
44
45
46
47
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Homework #20: Coin Word Problems
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Lesson #21: Consecutive Integer Word Problems
Do Now:
1] Simplify: (x
–
7)
2
2]
Solve:
6y – (6 + 4y) = 26
3]
The sum of two numbers is 84,
and one of them is 12 more than
the other. What are the two
numbers?
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Consecutive Integers
Find the pattern and fill in the blank:
Consecutive Integer
Expression:
2, 3,
______, 5
Consecutive
Even
Integer
Expression:
2,
4
, ______,
8
Consecutive Odd Integer
Expression:
9, 11, 13, _______
_______, -10 ,-8, -6
QUICK CHECK:
1] If x +3 represents an integer, then the next consecutive integer in terms of x is
(1) x (2) x +2 (3) x +4 (4) x +5
2] If x - 4 represents an odd integer, then the next consecutive odd integer in
terms of x is
(1) x - 6 (2) x - 4 (3) x - 3 (4) x - 2
What are
consecutive
numbers
?!?
Add 2 or
Add 1
?
Add 2 or
Add 1
?
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Steps for Solving ALL Word Problems:
1
st
Assign the variable
2
nd
Write an equation
3
rd
Solve
4
th
Check your solution
Consecutive Integer Example:
The sum of two consecutive integers is 15. Find the integers
Consecutive Integer Example:
The sum of three consecutive integers is 99. Find the three integers.
Consecutive Integer Example w/ Translation:
Find three consecutive numbers whose sum is 9 more than twice the
largest number.
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Consecutive Even & Odd Integer Examples:
Consecutive Even Example:
Find three consecutive even integers such that their sum is 42
Consecutive Odd Example:
Find three consecutive odd integers such that their sum is 57.
Consecutive Even Example w/ Translation:
Find three consecutive odd integers such that twice the sum of the
second and the third is 43 more than three times the first.
Remember!
n, n+2, n+4, . . .
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Tie-The-KnoTT:
Three sisters have ages that are consecutive odd integers. Find the
ages if the sum of the age of the youngest and three times the age of
the oldest is five less than five times the middle sister’s age.
U-Try:
1] Find two consecutive integers such that their sum is 89. Only an
algebraic solution will be accepted
2] Find three consecutive integers with sum 204.
3] Find three consecutive odd integers such that the sum of the first
and third equals the sum of the second and 31.
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HW #21: Consecutive Integer Word Problems
1] If x +2 represents an odd integer, then the next consecutive odd
integer in terms of x is
(1) x +1 (2) x +3 (3) x +4 (4) x +5
2] Find three consecutive integers with sum 168. Which is the
greatest of the three?
[A] 59 [B] 57 [C] 19 [D] 53
3] Find three consecutive integers with sum –99.
4] Find three consecutive odd integers such that the sum of the first
and third equals the sum of the second and 43.
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5] Find two consecutive integers such that the larger is nine more than
twice the smaller.
6] Three brothers have ages that are consecutive odd integers. The
difference between the age of the oldest and twice the age of the
youngest is twenty-four less than the middle brother’s age. Find their
ages
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Lesson #23: Solving and Graphing Inequalities on a
Number Line
Do Now:
1] Simplify:
2a – 3b –(b
2
– 4c +2a)
2] Solve:
2x + 3x – 4 = 11
3] Solve:
1 = y + 3(y – 9)
4] Simplify: (x – 3)
2
5] Solve for x:
3x
4
6 9
6] Simplify:
(3x
3
y
2
z
4
)
2
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Solving Linear Inequalities
An inequality is a statement that consists of two mathematical
expressions joined by an inequality symbol.
Fill in the blanks with the appropriate symbols: ( >, ≤, ≥, <, = )
1] 7 ___ 5 2] 13 ___ 16 3] -9 ___ -4
4] 17 ___ 17 5] -13 ___ -16 6] 0 ___ -3
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Graphing Inequalities on a Number Line:
Solving Linear Inequalities:
All the rules for solving equations apply to inequalities
When an inequality is multiplied or divided by ANY NEGATIVE
number, then the DIRECTION of the inequality sign CHANGES
Open Circle: Closed Circle:
Graph: x < 2
Graph: x ≤ 2
Graph: x > 3
Graph: x > 3
Exclusive
Vs.
Inclusive
To
solve
inequalities,
treat them like
EQUATIONS!
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The SOLUTION of an inequality includes ALL values that make the
Inequality TRUE = SOLUTION SET
Solve and Graph solution set on a number line for the following:
b + 7 ≤ 4
-4w < 20
3r - 17 ≥ 2r + 14
3(2x + 4) > 4x + 10
Symbol
How to Read It
Circle’s Appearance
greater than
open (not a solution)
less than
open (not a solution)
greater than or equal to
closed (is a solution)
less than or equal to
closed (is a solution)
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YOU TRY
1)
11 > 4x + 31
2)
12
–
⅔
x > 6
3)
3(5x + 7)
>
81
4)
3(4x + 1) <
-
27
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0 2 4 6 8 100–2–4–6–8–10
0 2 4 6 8 100–2–4–6–8–10
Writing An Inequality Based On A Graph
_________________
__________________
_________________
_________________
0 2 4 6 8 10 12 14 16 18 20 220
0 2 4 6 8 10 120–2–4–6–8
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HW #22: Solving and Graphing Inequalities on a Number
Line
Directions: Solve and graph the solution set for each of the following:
1] x – (-12) < 4
2] 2m + 7 > 17
3] 5(2h – 6) – 7(h + 7) > 4h
4] 12x – 17 > 19
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Solving Inequalities Practice Problems
64
65
Solving Inequalities Regents Test
66
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Lesson #23: Solving and Graphing Compund
Inequalities
Do Now:
Algebraic Symbols “Math Symbols”
Symbol
In WORDS
Graphed on # Line
Open vs. Closed
=
>
Greater Than
<
Less Than
≤
≥
≥
@ least
≤
@ most
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Compound Inequalities:
A compound inequality is two simple inequalities joined by "and" or "or".
Compound Inequalities:
Solving with “AND”
Solving with “OR”
Graph:
Graph:
Solution:
Solution:
Graph the solution set of each compound inequality:
1] x
>
-
3 and x
<
4
2] n
≤
-
5 or n
≥
-
1
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Solving Compound Inequalities:
Solve and graph solution set for the following:
1]
-
3
<
x + 1
≤
2
2]
-
4
<
-
6
x
+ 2
<
8
3] In the set of positive integers, what is the solution set of the
inequality
2x - 3 < 5?
A) {0, 1, 2, 3} B) {1, 2, 3} C) {0, 1, 2, 3, 4} D) {1, 2,
3, 4}
4] Which of the following is NOT a solution of - 4 < -6x + 2 < 8?
A) 0 B) 1 C) 2 D) -1
REMEMBER:
The SOLUTION(S) of
an INEQUALITY are
ALL the values that
make it
TRUE!
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Homework #24: Solving and Graphing Compound
Inequalities
Directions: Solve and graph the solution set for each of the following:
1] x – (-12) < 4
2] 2m + 7 > 17
3] -7 < 2x – 5 < 7
4] 5(2h – 6) – 7(h + 7) > 4h
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Inequality Word Problem Practice
Use each scenario to write an inequality that can be used to solve each situation:
1. R & G Catering specializes in catering wedding receptions. They charge
$550 for setting up the buffet and an additional $6.50 per guest. Mr. and
Mrs. Henderson want to spend no more than $1200 on their daughter’s
wedding reception. Write an inequality in terms of the number of guests, g
that they can invite to the wedding reception.
2. Anthony is carpeting several rooms in his home. The carpet costs $14.95 per
square yard plus $200 for installation. He can afford to spend no more than
$3000. Write an inequality to represent how many square yards of carpet
Anthony can afford.
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3.
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Inequality Regents Test
74
75
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