Algebra Notes

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Algebra Notes
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• Algebra contains formulas, variables,
  expressions, equations, and inequalities. All of
  these things help us to solve problems.

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• Variables are letters that represent numbers.
• Example n, x, and y they are the most commonly
  used variables.
• Algebraic Expression a combination of numbers,
  variables, and operations (x, , -, ).
• Example 2n 1
• Verbal Expression The meaning of an algebraic
  expression written out in words (directions).
• Example Two times a number n increased by one.

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• What is the difference between an expression
• and an equation?
• An equation has an equal sign and an expression
  does not.

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• Examples
• A number y increased by seven is twelve
• y 7 12
• The product of two and a number is equal to
  fourteen
• 2n 14
• Four less a number equals two times that number
• 4 n 2n
• Twice a number increased by six is ten less than
  a number
• 2n 6 n - 10
• Fourteen divided by a number increased by 6 is
  seven less than twice a number.
• 14/n 6 2n 7

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• y 3 9
• a number y increased by three is nine.
• 6n 17 3n
• Six times a number n minus seventeen equals three
  times that number.
• 4n 5 30/n
• The product of four and a number less five is the
  same as the quotient of thirty and that number
• 12 x 4 x
• Twelve divided by a number equals four more than
  the number
• 12x - 10 8 n
• Ten less than twelve times a number is eight less
  than that number

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Terms of an Expression

• Terms are parts of a math expression separated by
  addition or subtraction signs.

3x 5y 8 has 3 terms.
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Terms of an Expression

• Polynomial an expression that contains one or
  more terms.

Examples
Non-Examples
3x 2
4xy
2 x
3
x² 3x 4
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monomials
binomials
trinomials
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Monomials

• Monomial an expression that contains only ONE
  term.

Examples
Non-Examples
4xy
2x - 1
3
x² 3x - 4
x²
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Binomials

• Binomial an expression that contains exactly TWO
  terms.

Examples
Non-Examples
3x² y - 1
3x² x
y²- 7xy
3xyz
y - 5
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Trinomials

• Trinomial an expression that contains THREE
  terms.

Examples
Non-Examples
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TERMS NAME
1
2
3
4 or more
Monomial
Binomial
Trinomial
Polynomial
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Like Terms

• Like Terms have the same variables to the same
  powers

8x²2x²5a a 8x²and 2x² are like terms 5a and
a are like terms
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LIKE terms Yes or No?
3x and 7x
Yes - Like
5x and 5y
No - Unlike
x and x²
No - Unlike
Yes - Like
6 and 10
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Identify the LIKE terms
3m 2m 8 3m 6
5x b 3x 4 2x 1 3b
-6y 4yz 6x² 2yz 4y 2x² - 5
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Coefficients

• A Coefficient a number.

The coefficient is written in front of the
variable.
Example 6x
The coefficient is 6.
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Simplify

• Simplify means to combine like terms.

• You can simplify an expression combining like
  terms.
• Combine LIKE terms by adding their coefficients.
• You can ONLY combine LIKE TERMS
• You can NEVER combine UNLIKE TERMS

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Simplifying Rules
If there is no number written in front of a
variable, its coefficient is ONE. Example
x 1x When an expression has a subtraction
sign in front of it, the subtractions sign stays
with that term.
Example 3x 12 x 5 2x
3x 1x 2x 2x
12 5 7
2x 7
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Write an expression

3c 4c
7c
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Write an expression
-
8a - 1a
7a
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Write an expression

5c 4d
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Write an expression
-
5a 4b
This expression cannot be simplified. Why not?
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Simplify the following
3x 8x 2y
11x 2y
7x 3y 4 5x 2x
14x 3y -4
10x 3y 4x 5y
14x 2y
5z 7y 3x z y x
6z 8y 4x
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Simplify the following
4x 5y 3x - 2y
7x 3y
5x 3 4x - 12 4y
x 4y - 9
-3x 3 3y 5y y
7y 3x 3
3 4y 2x²- 7x 3xy 2xy 4
2x² xy -7x 4y 7
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The Distributive Property

• Distributive Property the process of
  distributing the number on the outside of the
  parentheses to each term in the inside.

a(b c) ab ac
Example 5(x 7) 5x 35
5x
57

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Practice 1 3(m - 4) 3 m - 3 4 3m
12 Practice 2 -2(y 3) -2 y (-2) 3 -2y
(-6) -2y - 6
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Simplify the following
3(x 6)
3x 18
4(4 y)
16 4y
18y 180
6(3y - 30)
5(2a 3)
10a 15
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Simplify the following
6(3y 5)
18y - 30
3 4(x 6)
4x 27
2x 3(5x - 3) 5
17x - 4
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REVIEW
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Which of the following is the simplified form of
5x - 4 - 7x 14 ?

1. -12x 10
2. -2x 10
3. 2x - 18
4. 12x 18

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Bonus! Which of the following is the simplified
form of a - 3a - 4(9 - a) ?

1. -36
2. 3a - 36
3. 2a - 36
4. 8a 36

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Which of the following is the simplified form of
(x 3) (x 4) ?

1. -2x 7
2. 2x - 1
3. 7
4. -1

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Which of the following is the simplified form of
-4x 7x ?

1. -4
2. 3x
3. -3x
4. 4

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Which statement demonstrates the distributive
property incorrectly?

1. 3(x y z) 3x 3y 3z
2. (a b) c ac bc
3. 5(2 3x) 10 3x
4. 6(3k - 4) 18k - 24

Answer Now
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Steps to Solving Equations

• Equation a mathematical sentence using an equal
  () sign.
• Step 1 Get rid of the 10. Look at the sign in
  front of the 10, since it is subtraction we need
  to use the opposite operation (addition) to
  cancel out the 10
• Add 10 to both sides. Remember, what you do to
  one side of the equation, you have to do to the
  other.

2n 10 50
10
10
2n 60
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Steps to Solving Equations

• Step 2 Next, we need to look at what else is
  happening to the variable. 2n means that two is
  being multiplied to n, therefore we need to do
  the opposite (division) to undo the
  multiplication.
• Divide both sides by 2. Remember, what you do to
  one side of the equation, you have to do to the
  other.

2n 60
2
2
n
30
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Steps to Solving Equations

• Step 3 Plug Chug then CHECK your solution!!
  First, rewrite the original equation
• We already solved for n, so wherever you see the
  variable, n, plug in the answer.
• Evaluate the equation, SHOWING ALL WORK!
• Does it check?

2n 10 50
2 (30) 10 50
60 10 50
50 50
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Solve Check

• 5n 15 100
• 105 10n 5
• n/5 3 6
• -44 7n 250
• 200 100 25n
• 12 4n 112

n 23
n 10
n 15
n 42
n -4
n 25
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Steps to Solving Multi-Step Equations

• Step 1 Distribute if necessary variable.
• Distribute the 4 to the n and 5.

4(n 5) - 7 9 2n 4n
4n 20 - 7 9 2n 4n
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Steps to Solving Multi-Step Equations

• Step 2 Combine like terms on each side of the
  equations.
• On the left side -20 and -7 combine to get -27
• On the right side 2n and -4n combine to get -2n

4n 20 - 7 9 2n 4n
4n 27 9 2n
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Steps to Solving Multi-Step Equations

• Step 3 Get all variables to one side of the
  equation.
• First we want to get rid of the -27. Look at the
  sign in front of -27, since it is subtraction (or
  a negative) we need to use the opposite operation
  (addition) to cancel it out. Therefore add 27 to
  both sides.

4n 27 9 2n
27
27
4n 36 2n
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Steps to Solving Equations

• Step 4 Get all plain numbers to one side of
  the equation
• First we want to get rid of the -2n. Look at the
  sign in front of -2n, since it is subtraction (or
  a negative) we need to use the opposite operation
  (addition) to cancel it out. Therefore add 2n to
  both sides.

4n 36 2n
2n
2n
6n 36
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Steps to Solving Multi-Step Equations

• Step 5 Next, since we have all the variables on
  one side and all the plain numbers on the other
  side we need to look at what else is happening to
  the variable.
• 6n means the 6 is being multiplied by n,
  therefore we need to do the opposite (division)
  to undo the multiplication. So, divide both
  sides by 6.

6n 36
6
6
n 6
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Steps to Solving Multi-Step Equations

• Step 6 Plug Chug then CHECK your solution!!
  First, rewrite the original equation
• We already solved for n, so wherever you see the
  variable, n, plug in the answer.
• Evaluate the equation, SHOWING ALL WORK!
• Does it check?

4(n 5) - 7 9 2n 4n
4(6 5) - 7 9 2(6) 4(6)
4(1) - 7 9 12 24
4 7 21 - 24
-3 -3
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Solve Check

• 9 5r -17 8r
• 3(n 5) 2 26
• 58 3y -4y 19
• 4 2(v 6) -8
• 5(y 2) 6 6y 2y 14 y

r -2
n 3
y -11
v 12
y 5
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LESCA

• An electrician charges 50 to come to your
  house. Then he charges 25 for each hour he
  spends there. If the electrician charges you a
  total of 125, how many hours did he spend there?
• Let Statement Let number of hours x
• Equation 50 25x 125
• 25x 75
• Solution x 3
• Check 50 25(3) 125
• 50 75 125
• 125 125
• Answer Sentence The electrician was there for 3
  hours

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LESCA

• The sum of two consecutive integers is 73. What
  are the numbers?
• Let Statement Let first number x
• second number x 1
• Equation x x 1 73
• 2x 72
• Solution x 36
• Check 36 36 1 73
• 73 73
• Answer Sentence The numbers are 36 and 37.

36
37
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LESCA

• A taxi charges 1.50 plus a fee of 0.60 for
  each mile traveled. If a ride costs 5.40, how
  many miles was the ride?
• Let Statement Let number of miles x
• Equation 1.50 0.60x 5.40
• .6x 3.6
• Solution x 6
• Check 1.50 0.60(6) 5.40
• 1.5 3.6 5.40
• 5.4 5.4
• Answer Sentence The ride was 6 miles.

6
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LESCA

• Two years of internet service costs 685,
  including the installation fee of 85. What is
  the monthly fee?
• Let Statement Let monthly fee x
• Equation 24x 85 685
• 24x 600
• Solution x 25
• Check 24(25) 85 685
• 600 85 685
• 685 685
• Answer Sentence The monthly fee is 25.

25
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LESCA

• The sum of two numbers is 99. The difference of
  the two numbers is 9. What are the numbers?
• Let Statement Let first number x
• second number x 9
• Equation x x - 9 99
• 2x 108
• Solution x 54
• Check 36 36 1 73
• 73 73
• Answer Sentence The numbers are 45 and 54.

54
45
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Inequalities

• Inequality a mathematical sentence using lt, gt,
  , or .
• Example 3 y gt 8.
• Inequalities use symbols like lt and gt which
  means less than or greater than.
• They also use the symbols and which means
  less than or equal to and greater than or equal
  to.

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Whats the difference?

• x lt 4 means that x is less than 4
• 4 is not part of the solution
• What number is in this solution set?
• x 4 means that x can be less than OR equal to 4
• 4 IS part of the answer
• What number is in this solution set?

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You graph your inequalities on a number line

• This graph shows the inequality x lt 4
• The open circle on 4 means thats where the graph
  starts, but 4 is NOT part of the graph.
• The shaded line and arrow represent all the
  numbers less than 4.

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What is this inequality?

• X gt -2

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What is this inequality?

• X 2 1/2

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Graphing inequality solution sets on a number
line

• Use an open circle ( ) to graph inequalities
  with lt or gt signs.
• Use a closed circle ( ) to graph
  inequalities with or signs.

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What do you think this symbol means?
?

• Does not equal
• Example x ? 7

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Graph x ? -1

• X ? -1 would include everything on the number
  line EXCEPT -1.
• Use an open circle to show that -1 is NOT a part
  of the graph.

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Solve, Graph, Check

• 2n 7 gt 13
• 3y 9 -3
• 2n 10 6
• 5n 4 lt 4n
• 3x 3 9
• 4 7x lt 25

n gt 3
y 2
n -2
n lt -4
x 4
x lt 3
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Evaluating Formulas

• Evaluate means to replace variables with their
  numerical values and then solve
• Example n 3 5 n 2
• Example y 3, if y 9
• Then, y 3 9 3 6

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