ALGEBRA UNIT 3
SOLVING LINEAR EQUATIONS AND INEQUALITIES
PROPERTIES/PEMDAS (DAY 1)
Property Name
Basic Definitions
(how to remember properties)
Examples
a+(b+c)=(a+b)+c
3rd Man out ________________
or
a(bc)=(ab)c
3rd Man out ____________
a+(b+c)=a+(c+b)
Or
____________ ___________
or
Get Rid of __________________
a+b=b+a
3(a + b) = 3a + 3b
Divide out___________ element
Rewrite expression with
ax + bx = x(a + b)
__________________
______ _______________ value
to = ZER0 (0)
3 + -3 = 0
4 
__________ by __________ value
1
1
4
To = ONE (1)
Add “what” to your value to
get back your value
a+0=a
Multiply by “what” to your
value to get back your value
7  17
Ex1:
Justify the steps of the following simplified expression using properties
3(x  4)  5(x  2)
3 x  12  5x  10
3 x  5x  12  10
 2 x  22
Ex2:
GIVEN
_______________________________
_______________________________
______________________________
Simplify the following expression and justify each step.
Ex 3. Identify the property illustrate below:
a)
7 • ( 3 • 5) = (7 • 3 ) • 5
____________
b)
3( a + 4) = 3a + 12
____________
c)
c  -c  0
____________
d)
1•17 = 17
____________
e)
8•4 = 4•8
____________
f)
3+0 = 3
____________
g)
1
2( ) = 1
2
____________
h)
6x  6y  6(x  y)
____________
i)
9 •1= 9
____________
j)
____________
4(ab)  4(ba)
Ex4. Fill in the blank with the missing element. Name the property represented by the sentence.
a) 6  13  13  _____
____________
e) 8 + ___ = 8
____________
b) 5(3 + 7) = 5(_____) + 5(7)
____________
f) (___)6 = 6
____________
c) 3  (5  8)  (3  5)  _____
2 4 3
4 3
d)      _____    
3 9 8
9 8
____________
____________
g) -9 + _____ = 0 ____________
h)
3
(____)  1
4
____________
Ex5. Calculator Practice: Use your calculator to evaluate the following:
30
=
2
__________
b) 2 + [48 ÷ (12 + 4)] = __________
e) 6  5[4  3(2  1)] =
__________
c) 3 • 2[4 + (9 ÷ 3)] =
f) 50 ÷ [(4 • 5) - (36 ÷ 2)] =
__________
a) [45 – (3 • 2)] ÷ 3 =
__________
__________
d) [5(20 – 2 )] ÷
SOLVING LINEAR EQUATIONS (DAY 2)
How to Solve Basic Equation:
7x + 5 = 40
-5 -5
7x = 35
7 7
x=5
Box VARIABLE TERM and Solve for variable


Get BOXED TERM by itself on one side of the = sign
Solve for VARIABLE BOXED TERM:
How to Solve Equation with ( )
 Circle # in front of ( )
**if a fraction follow directions below for fractions in equation.**
 Use distributive property to get rid of
 Solve remaining equation using steps from above
3y – (8y – 6) = 46
3y – 8y + 6 = 46
-5y + 6 = 46
-6 -6
-5y = 40
-5 -5
y = -8
Solve the following:
1.
3.
5.
4x + 3 = 11
2.
8y – (5y + 2) = 16
9x – 6 = 5x – 15 + x
4.
3(x  4)  2  16
–(-x + 3) = 5x + 5 – 4x
6.
4(3x – 1) + 8 = 2(5x + 2) + 2x
3
HOW TO SOLVE EQUATIONS WITH FRACTIONS
Type 1: Fraction with NO ( ) in problem

*6
*6
*6 5
x  17  102
6
Multiply EVERY term by the bottom # to
get rid of the fraction
5x  102  612
+102 +102
5x = 714
5
5
x = 142.8
This will cancel the Fraction out

Then solve basic equation
Type 2: Fraction with ( ) in problem

*4
*4 3
(3x  8)  15
4
Multiply both sides by bottom # to get rid
of the fraction
This will cancel the Fraction out
3(3 x  8)  60
Distribute remaining # in front of ( )
Then solve basic equation
9x  24  60
-24 -24
9x = 36
9 9
x=4


Solve the following:
7.
3
(3x  8)  15
4
8.
2
x  19  7
3
9.
3x
 7x  7  3(2x  1)
2
10.
5x 
4
2
(4x  3)  20
9
SOLVING EQUATIONS CONT…(DAY 3)
Solve the following for the variable.
1.
4(3 x  5)  x   x  16
3.
2x 
5.
 36(7  3 x)  6(2  2 x)
7.
x x
 7
3 4
5
5
x
8
8
5
2.
5x  14  5  8x
4.
10(2  x)  5(1 3 x)  30
6.
1
7
5x  (3x  8)  4  x
2
2
8.
6x 45

5
3
SOLVING LINEAR INEQUALITIES EQUATION ( , , , ) (DAY 4)
PROCEDURE FOR SOLVING INEQUALITY SOLUTION SETS

Solve the inequality equation
Inequality Rule:
When you ____________ by a ___________ number, you MUST _______ the inequality sign.

Graph the solution on a ______________________, using correct endpoints
________/__________circle.
RECALL:

< or >
have __________ circles
 or 
have __________ circles
Shade where solutions make inequality statement _____________. Pick a
_____________________ to determine truth value.
Example: Solve and graph the inequality below, write the solution set in interval notation
- x  6  - (2 x  4)
HOW TO WRITE SOLUTION SET AFTER INEQUALITY WAS GRAPHED
There are ____ different ways to notate the set of answers for an inequality equation
Interval Notation:
Use round parenthesis/square brackets and a comma between endpoint
values on number line
Shading that doesn’t stop is headed towards ______ or _______. ______ _________________ ALWAYS!
Set Notation: Use inequality symbols (<, >,  ,  ) and
6 the variable x.
Solve and graph the following inequalities. Write the solution set using Interval Notation
1.
2
x  1  15
3
2.
3.
1
x  5  x – 9
5
4.
5.
Write the solution set of the following using Interval Notation
a.

c.
o
d.
-7 -6 -5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
b.

12  2x  6
4 + 9y – 3  3(3y + 2)
o
o
-7 -6 -5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
o

-7 -6 -5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
-7 -6 -5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
7
SOLVING APPLICATION WORD PROBLEMS (DAY 5)
1.
A dance academy charges $24 per class and a one-time registration fee of $15. A
student paid a total of $687 to the academy. Find the number of classes the student
took.
2.
Tyler paid $124 to get his car repaired. The total cost for the repairs was the sum of the
amount paid for parts and the amount paid for labor. Tyler was charged $76 for parts
and $32 per hour of labor. Write and solve an equation to determine the amount of
time it took to repair his car.
3.
On opening day at Darien Lake, Kelly pays $40 for admission to the park. She wants to
play the games in the midway. Each game cost $1.50 to play. If she doesn’t want to
spend more than $65, how many rides or games could she participant on?
4.
As a waiter at a restaurant, Joe earns $2.25 an hour plus tips. If he made $55 in tips
and his total earnings did not exceed $70, how many hours could he have worked?
5.
Leonard wants to save $100 in the next two months. He knows that in the second
month he will be able to save $20 more than during the first month. How much should
he save each month?
8
APPLICATION WORD PROBLEMS CONT… (DAY 6)
1.
Mike started a lawn-mowing business for the summer. He bought a used lawn mower
for $75. He plans on charging $15 per lawn in his neighborhood. At the end of the
summer Mike wants to go on a vacation before school starts, which costs $650. What is
the minimum number of lawns he needs to mow to be able to go on his trip?
Mike forgot to figure out the cost of gas. Since he estimated that he would use 1
gallon of gas for every 2 yards mowed. The cost of gas was $3.89 per gallon. How
many more lawns does he need to mow to still be able to go on the trip?
9
CONSECUTIVE INTEGER WORD PROBLEMS (DAY 7)
Integers:
Consecutive Integers:__________________________________________
Let statements:
Consecutive Even Integers:__________________________________
Let Statements:
Consecutive Odd Integers:__________________________________
1.
Find two consecutive even integers such that three times the smaller is 30 more than
twice the larger.
2.
Find two consecutive even integers such that 4 times the larger exceeds 3 times the
smaller by 2.
3.
The length of the shortest side of a right triangle is 8 inches. The lengths of the other
two sides are represented by consecutive odd integers. Which equation could be
used to find the lengths of the other two sides of the triangle?
(1)
(2)
(3)
(4)
8 2  (x  1)  x 2
x 2  8 2  (x  1)2
8 2  (x  2 )2  x 2
x 2  8 2  (x  2)2
10
4.
The sides of a pentagon are represents by consecutive even integers. The perimeter
of the pentagon is 50 ft. Find the length of each side.
5.
The base of an isosceles triangle and one of its legs has lengths that are consecutive
integers. The leg is longer than the base. The perimeter of the triangle is 20. Find the
length of each side of the triangle.
6.
The Smith family has four children. Their ages are represented by consecutive integers.
If the sum of the first three exceeds twice the oldest by 4, determine the ages of all the
children.
11
APPLICATION WORD PROBLEMS (DAY 8)
1.
Dave, Steven, and Becky met up at Frostys to get some ice cream, but they only had
nickels, dimes and quarters on them. Dave had quarters, Steven had nickels, and
Becky had dimes. Becky has two more than twice number of coins than Dave, and
Steven has 8 more coins than Dave. If they have exactly enough for three double
scoop ice cream cones, which cost $6.60, how many quarters, dimes, and nickels do
they have?
2.
Erin wants to make trail mix made up of almonds, walnuts, and raisins. She wants to
mix one part almonds, two parts walnuts, and three parts raisins. Almonds cost $12 per
pound, walnuts cost $9 per pound, and raisins cost $5 per pound
Erin has $15 to spend on the trail mix. Determine how many pounds of trail mix she can
make.
12
3.
At a local high school football game, $1,350 was collected for hot dogs, hamburgers,
nachos, and soft drinks. Three times as many hot dogs were sold as hamburgers and
twice as many nachos were sold as hamburgers. As many soft drinks were sold as hot
dogs, hamburgers, and nachos combined. If each item sold for $1.50, how many soft
drinks were sold?
4.
Irene has $4.50 in her coin bank in nickels and dimes. The number of dimes is twice the
number of nickels. How many coins of each type does Irene have?
5.
Triton Cell Phone Services offers a great deal for new customers. Each month, the
customers pay a flat fee of $45, plus $.23 per minute for international phone calls.
Local and domestic long distance calls are unlimited and included in the flat fee.
Tricia called her aunt in France and spoke to her for 35 minutes. How much would her
cell phone bill be for that month if she used Triton?
13
LITERAL EQUATIONS (DAY 9)
1.
Given the equation c = k + 273 solve for k in terms of c.
2.
If 3ax  b  c , solve for x in terms of a, b and c. Identify the properties used.
3.
Solve the following equation for x. Identify the properties used.
ax  bx  c .
1
Bh , solve for B in terms of V and h
3
4.
Given the formula V =
5.
If the formula for the perimeter of a rectangle is P = 2(L +w), then w can be expressed
as
(1) w 
6.
2L  p
2
(2) w 
p  2L
2
(3) w 
Solve the literal equation a(x+b) = c for x.
14
p L
2
(4) w 
p  2w
2L
In #7-9 solve for y in terms of x:
7.
9x + y = 15
8.
-5x + 2y = 8
10.
If 2ax – 5x = 2, then x is equivalent to
2  5a
2a
1
(2)
a5
(1)
(3)
9.
12 = 6x – 3y
2
2a  5
(4) 7  2a
11.
The Freshmen class is planning a dance. They use the equation r = pn to determine
the total receipts. What is n expressed in r and p?
12.
The perimeter P of a rectangle is given by the formula P  2 L  2W , where L is the
length and w is the width.
Solve the formula for the width w.
Use the rewritten formula found above to find the width of the rectangle shown.
P=19.5 ft
7.2 ft
15
w