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Algebra 1: NAME: __________________________
NOTES 1.1: Variables and Expressions
Definitions
Variable
Constant
Numerical Expression
Algebraic Expression
Mathematical Operations and Words
Example 1:
Give two ways to write each algebraic expression in words
x + 3 m – 7 2∙y k 5
You Try:
Give two ways to write each algebraic expression in words
4 – n 9 + q 3(h)
Example 2:
Translating from words to algebraic symbols
1) Eve reads 25 pages per hour. Write an expression for the number of pages she reads in h hours
2) Sam is 2 years younger than Sue, who is y years old. Write an expression for Sam’s age
3) William runs a mile in 12 minutes. Write an expression for the number of miles that William runs in m minutes.
You Try:
Translating from words to algebraic symbols
1) Lou drives 65 mi/h. Write an expression for the number of miles that Lou drives in t hours
2) Miriam is 5 cm taller than her sister, who is m cm tall. Write an expression for Miriam’s height in centimeters
3) Elaine earns $32 per day. Write an expression for the amount that she earns in d days
5
t
Definitions
Evaluating an Expression
Value of an Expression
Steps for Evaluating an Expression
STEP 1 STEP 2 STEP 3
Example 3:
Evaluate each expression for x = 8, y = 5, and z = 4
x + y
You Try:
Evaluate each expression for m = 3, n = 2, and p = 9
mn p – n p m
z
x
Example 4:
Approximately fourteen 20-ounce plastic drink bottles must be recycled to produce 1 square foot of carpet
◦ Write an expression for the number of bottles needed to make c square feet of carpet
◦ Find the number of bottles needed to make 40, 120, and 224 square feet of carpet
You Try:
To make one sweater, sixty-three 20-ounce plastic drink bottles must be recycled
◦ Write an expression for the number of bottles needed to make s sweaters
◦ Find the number of bottles needed to make 12, 25, and 50 sweaters
1.2 and 1.3 Note-Sheet: One – Step Equations
(Addition, Subtraction, Multiplication, and Division)
GOAL: to isolate the variable
What you do to one side of the
equal sign you must do to the
EXACT same thing to the
other side of the equal sign
Equations must balance
THINK OF OPPOSITES
Subtraction – Addition
Addition – Subtraction
Multiplication – Division
Division -- Multiplication
Inverse Operations
To solve one step equations, you need to
ask three questions about the equation:
• What is the variable?
• What operation is performed on
the variable?
• What is the inverse operation?
(The one that will undo what is
being done to the variable)
Solve using the inverse operation
Helpful Hints
* Work on the side with the variable
* Get rid of ALL fractions first by
multiplying EVERY term by the
denominator
* You can check your solution by
substituting your answer back into the
original problem and evaluating (both
sides of the equal sign should be the
same if your solution is correct)
* Other (Do you have any hints)?
Example 1:
x – 10 = 4
1) What is the variable? ___________
2) What operation is being performed on the variable? ____________________
3) What is the inverse operation? ______________________________
Solve showing ALL work!!!
x – 10 = 4
You Try:
-5 = k + 5
1) What is the variable? ___________
2) What operation is being performed on the variable? ____________________
3) What is the inverse operation? ______________________________
Solve showing ALL work!!!
-5 = k + 5
Example 2:
7x = 56
1) What is the variable? ___________
2) What operation is being performed on the variable? ____________________
3) What is the inverse operation? ______________________________
Solve showing ALL work!!!
7x = 56
You Try:
54
k
1) What is the variable? ___________
2) What operation is being performed on the variable? ____________________
3) What is the inverse operation? ______________________________
Solve showing ALL work!!!
54
k
Example 3:
359
5v
Solve showing ALL work!!!
359
5v
You Try:
r8
39
Solve showing ALL work!!!
r8
39
Example 4:
A person’s maximum heart rate is the highest rate, in beats per minute, that the person’s heart should reach. One method to estimate
maximum heart rate states that your age added to your maximum heart rate is 220. Using this method, write and solve an equation to find the
maximum heart rate of a 15-year-old
Define the variable
r = maximum heart rate
Key Phrases in the problem and what they mean.
1) age added to maximum heart rate:
2) is 220:
3) age:
Equation:
Solution:
You Try:
Over 20 years, the population of a town decreased by 275 people to a population of 850. Write and solve and equation to find the original
population.
Define the variable = _______________________________________
Key Phrases in the problem and what they mean.
Equation:
Solution:
What??? I just learned 1-step!
Relax. You’ll use what you already know
to solve 2-step equations.
1.4 Note-Sheet: SOLVING EQUATIONS
(Notes: 3 parts: two-step equations, fractions, distribute and combine like terms)
Two – Step Equations
AND
Equations With Fractions
Two – Step Equations
STEPS: Follow all steps below to solve two-step equations (GOAL: get the variable by itself)
1st = Work on the side with the variable. Add or subtract (this is the number without the variable)
2nd = Divide last (this is the number with the variable)
Example 1 Steps Taken
5q – 7 = 13
Example 2 Steps Taken You Try Steps Taken
8x + 5 = 61 3a – 8 = 4
Example 3 Steps Taken You Try Steps Taken
-47 = 3x – 50 -35 = 2p + 10
Equations with FRACTIONS!!!
STEPS: Follow all steps below to solve equations with fractions (GOAL: get the variable by itself)
1st = If you see fractions; get RID of them FIRST by multiplying EVERY term by the denominator
2nd = Work on the side with the variable. Add or subtract (this is the number without the variable)
3rd = Divide last (this is the number with the variable)
Example 1 Steps Taken Example 2 Steps Taken
208
5y 157
3
2
x
You Try Steps Taken
145
10 m
EQUATIONS:
Distributive Property and Combining Like Terms
STEPS: When ALL variables are on the same-side of the equal sign.
1st = If you see parenthesis ( ) then distribute first
2nd = Combine any like terms on the same-side of the equal sign
3rd = Work on the side with the variable. Add or Subtract (this is the number without the
variable)
4th = Divide last (this is the number with the variable)
EXAMPLE #1 12 – x = -5
GOAL: Get the "x" by itself
You TRY. . .
12 – x = -5 Steps Taken 21 = -x – 3 Steps Taken
EXAMPLE #2 4x + 2 + 3x = 58
GOAL: Get the "x" by itself
You TRY. . .
4x + 2 + 3x = 58 Steps Taken 10x + 2 – 5x = 52 Steps Taken
EXAMPLE #3 4(x + 12) + 7x = 26
You TRY. . .
4(x + 12) + 7x = 26 Steps Taken -2(x – 3) = -12 Steps Taken
EXAMPLE #4
The price of an automobile tire has been reduced by $15. The cost of a set of 4 tires at the reduced price is
$600. Write and solve an equation to find the original cost of a tire.
Define the variable.
X = cost of tire
Key phrases in the problem and what they mean.
1) “reduced” by $15 =
2) “set of 4 tires” =
3) “the reduced price is $600” =
4) Equation: _______________________________________
5) Solution = ________________________________________
You Try
The height of an ostrich is 20 inches more than 4 times the height of a kiwi. Write and solve an equation to find the
height of a kiwi.
Define the variable
Identify key phrases in the problem and what they mean
Equation
Solution
1.5 Note-Sheet: Variables on Both Sides:
STEPS: Follow all steps below to solve equations
1st = If you see parenthesis ( ) then distribute first
2nd = Combine any like terms on the same-side of the equal sign
3rd = Add or subtract to move variable terms to one side of the equal sign (Variables on
one side, numbers on the other side)
4th = Work on the side with the variable. Add or Subtract (this is the number without
the variable)
5th = Divide last (this is the number with the variable)
EXAMPLE #1 You Try #1
4x = 2x + 6 Steps Taken -3x + 15 = 2x Steps Taken
EXAMPLE #2 You Try #2
-4a + 6 = 2a – 36 Steps Taken 10w + 300 = 20w – 50 Steps Taken
EXAMPLE #3 You Try #3
-4(x – 4) = 6x + 36 Steps Taken 10x – 2 = 2(x + 7) Steps Taken
1.7 Note-Sheet: Solving Absolute-Value Equations
Absolute Value (of x)
Symbol |x|
The distance x is from 0 on a number line
Always positive
STEPS for Solving
1st = Get the | | symbol by itself (isolate it)
2nd = Set up two cases (one positive and one negative) NOTE: get rid of the | |
|x| = a
Case 1: x = a Case 2: x = -a
3rd = Solve each case
***HINTS:
1) 0 does not have and opposite (|0| = 0)
2) | | ≠ negative number (|x| ≠ -2)
Example: |x| = 5
What are the possible values of x?
Set up 2 cases and solve
Case 1 Case 2
You Try: |x| = 4
What are the possible values of x?
Set up 2 cases and solve
Case 1 Case 2
EXAMPLE: 4|x + 2| = 24 You Try: 2|x – 1| = 4
Case 1 Case 2 Case 1 Case 2
EXAMPLE: |x + 3| + 4 = 4 You Try: 5 = |x + 2| + 8
Case 1 Case 2 Case 1 Case 2