Algebra Notes

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Algebra Notes
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Writing Algebraic Expressions
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• Let Statement math sentence used to define a
  variable to represent the unknown quantities.

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• Laura has twice as much homework as Ann.
• The Bills won five more games than they lost.
• Seven more than three times a number is 25.
• The length of a rectangle is 3 cm more than the
  width.

Let Ann a Let Laura 2a
Let games lost g Let Bills won 5 g
Let Yankees y Let Tigers 3y
Let width w Let length 3 w
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• Mike is three years older than Jim.
• Eight more than twice a number is 32
• Seven more than three times a number is 25.
• Twice a number increased by four is 16.

Let Jim j Let Mike 3 j
Let number n 2n 8 32
Let number n 3n 7
Let number n 2n 4 16
6

• Six less than three times a number is 21.
• Fifteen less than twice a number is 25.
• Sixty-six is eleven more than five times a
  number.

Let number n 3n 6 21
Let number n 2n 15 25
Let number n 66 5n 66
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Writing Algebraic Expressions
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Setting up Solving word problems
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• Write your let statement
• Write your equation
• Solve
• Check
• Write an answer sentence

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• A cell phone company charges 39 a month plus
  .15 per text message sent. If Jan sends 35 text
  messages this month, how much does she owe before
  taxes are added?
• The Bills won five more games than they lost.

Let text message t
Jam owes 44.25
39 0.15t t 44.25
Let text message s
12 2s s 4
4 snacks
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• A rental car company ABC charges 25 per day plus
  .15 per mile. Rental car company XYZ charges 18
  per day plus .25 per mile. If you plan to drive
  50 miles, who is the cheaper rental company?
• Joe attends a carnival. The admission is 48.
  Tickets for rides cost 4 each. Joe needs one
  ticket for each ride. Write an equation Joe can
  use to determine the number of ride tickets, r,
  he can buy if he has 200 before he pays the
  admission fee.

Let miles m
ABC 25 0.15m 32.50
XYZ 18 0.25 30.50
XYZ is cheaper
Let number or rides r
48 4r 200 r 38
38 rides
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substitutions
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Evaluate if s 4

• 4s ?
• 4 s ?
• 5 s ?
• 12 s ?

4(4) 16
4 4 8
5 - 4 1
12 4 3
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Evaluate if s -6

• 7s ?
• 3 s ?
• 7 s ?
• 18 s ?

7(-6) -42
3 (-6) -3
7 (-6) 13
18 (-6) -3
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Evaluate if n 3 and r 5

• n² 7r
• 9n - r²
• 2nr 6n

3² 7(5) 9 12 21
9(3) - 5² 27 25 2
2(3)(5) 6(3) 30 18 48
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Evaluate if p 12 and q -8

• p q 6
• p q 3
• p q q²

12 (-8) 6 -4 6 2
12 (-8) 3 20 3 23
12 (-8) (-8)² 20 64 84
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Evaluate if a -2 and b 6

• 3a² 5b²
• 4a³ 3b
• 7a² - (b²/3)

3(-2)² 5(6)² 3(4) 5(36) 192
4(-2)² 3(6) 4(-8) 18 -14
7(-2)² - (6²/3) 7(4) (36/3) 28 12 16
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Like Terms
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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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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 7x
Yes - Like
5x 5y
No - Unlike
4c c
Yes - Like
No - Unlike
4d 4
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LIKE terms Yes or No?
3ab 6b
No - Unlike
Yes - Like
2a 5a
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 written in front of the
  variable.

Example 6x
The coefficient is 6.
Example x
The coefficient is 1.
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Simplify

• Simplify means to combine like terms.

• Combine LIKE terms by adding their coefficients.

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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 expressions
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• 2x 4x
• 2a 5a 6
• 3xy xy 2x
• -4c 8c 6c

• 3a 7a
• 3½y 5y -4y
• cd 4cd 2a
• ½e 2e ¾ e

• 6xy 2xy
• 5d 6d 3d
• 4s 4s
• 5x 4x 4x 11x

6x
10a
4xy
4½y
7a 6
-4d
2xy 2x
5cd 2a
0
24x
-¾e
-2c
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Challenge questions
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1. 5x 3x 2. 8x 2x 3. 7x
(3x) 4. 6x (4x) 5. 10x 14x 6. 9x
(x) 7. 3x 8x 8. x (5x) 9. a² b²
2a² 5b² 10. 7h² 3 2h² 4
-8x
6x
-4x
10x
-8x
-24x
-5x
6x
5h² 7
3a 6b²
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11. 3x 3y x y z 12. 5b 5b 6b² - 10
3b 13. Find the perimeter of the
rectangle A 4x 3y B 8x 6y C 12xy D
4x² 3y²
4x 4y z
6a² 7b - 10
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Adding subtracting Polynomials
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Adding

• Combine like term
• Add the coefficients to simplify
• Example Add   2x² 6x 5   and  3x² - 2x 1
• Start with 2x² 6x 5         3x² - 2x 1
• Place like terms together _______ ________
  ________
• Add the like terms _________ __________
  _________
• Final answer

6x 2x
5 1
2x² - 3x²
4x
5x²
4
5x² 4x 4
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Subtracting

• Change the subtraction sign to addition and
  reverse the sign of each term that follows
• Then add as usual
• Example Subtract   5y² 2xy - 5   and  3x² - 2x
  1
• Start with 5y² 2xy - 5     -     2y² - 3xy 3
• Place like terms together _______ ________
  ________
• Add the like terms _________ __________
  _________
• Final answer

-
-


2xy 3xy
-5 3
5y² - 2y²
-xy
3y²
-2
3y² - xy - 2
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Try the following
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1. (2x 3y) (4x 9y) 2.(3a 5b 7c) -
(5a 2b 9c) 3. (3x 5) (x 7) (7x
12) 4.(3a 5b 7c) (8a 2b
9c) 5. 4x³ 6x² 8x 10 and 7x³ 4x² 9x
3 6. Subtract (5m 6n 12) from (2m 3n
5). (2m 3n 5) - (5m 6n 12)
-2a 7b 2c
6x 12y
11x
11a 3b 2c
3x³ 2x² - x - 7
-3m 9n -17
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7.Subtract 8a 5b 6c from 10a 8b 7c (10a
8b 7c) - (8a 5b 6c) 8. (4x 8y 9z
7a 5b) (4b 5x 7y 3z 2a) 9. ( 3x2
4x 11) (6x2 8x 10) . 10. (7e² 3e
2) (9 6e 4e²) (9e 2 6e²)
2a 3b 13c
-x y 6z 9a b
3x² 12x - 21
5e² 6e 13
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challenge
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Some of the measures of the polygons are given.
P represents the measure of the perimeter. Find
the measure of the other side or sides.
x² - 15x 3
2x y
4x - 3
14x² - 4x 7
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The distributive property
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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
7(2 z)
14 7z
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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Distributive practice
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• 2(4 9x) 2. 7(x -1)
  3. 12(a b c)
• 7(a c b) 5. -10(3 2 7x) 6.
  -1(3w 3x -2z)
• -1(x 2) 8. 3(-2 2x2y3 3y2)
  9. 5(5 5x)
• y(1 x) 11. 12x(3x 3) 12.
  9(9x 9y)

8 18x
7x - 7
12a 12b 12c
-3w 3x 2z
7a 7c 7b
-70x - 50
-x 2
25x 25
-6 6x²y³ 9y²
y yx
36x² 36x
81x 8y
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Factoring
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factoring

• To factor expressions find the GCF (greatest
  common factor) of the terms
• Factoring is the opposite of distributing.

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Find the GCF of each pair of monomials

• 4x, 12x 2. 18a, 20ab
  3. 12cd, 36cd

12cd
4x
2a
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Factor each expression
4. 12a 6h 5. 3x 9
6. 12x y 7. 24a 4
8. 72a 9n 9. 8a - 8v
3(x 3)
6(2a h)
Cannot be simplified
4(6a 1)
9(8a n)
8(a v)
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Solving equations
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Steps to Solving Equations

• Equation a mathematical sentence that uses 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 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

• 105 10n 5
• n/5 3 6
• -44 7n 250
• -1/2 -5/18h
• 200 100 25n
• -9.4 z -3.6

n 10
n 15
n 42
h -9/5
n -4
z 5.8
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Solving equations practice
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• x 3 19 2. a 14 6
• 3. 9x 63 4. 5x 2 8
• 6. 8a 5 53
• -7 c 6 8. a 3.5 4.9
• 9. x 2.8 9.5 10. 2.25 b 1
• .

x 22
a 22
x 22
x 22
x -30
a 22
c 22
a 22
x 12.3
b 22
14. 2(b 2) b 3
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• 11. 12. -8.5 r -2.1
• 13. 14. 2(b 2) b 6.5
• .

r 6.4
c 1 3/7
m 33/14
b 2.5
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Solving multi-step equations
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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 multi-step 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 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

r -2
n 3
y -11
v 12
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Inequalities
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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 means
7 is not equal to x
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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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Graph x lt 4 (a number less than 4)
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Graph x lt 6 (a number less or equal to 6)
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Graph x gt 3 (a number less or equal to 6)
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Graph each inequality
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Graph

• x lt 3
• x gt -5
• x lt -1
• x gt 2

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Solve, Graph, and check each inequality
85
Solve, Graph, Check

• x 8 gt 15
• 3y 4 lt 11
• 2x lt 18
• x 4 gt 2
• 2n 7 gt 13

x gt 7
y lt 5
x lt 9
x gt -2
n gt 3
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Solve, Graph, Check

• 5n 4 lt 4n
• 3x 3 9

x lt -6
y 3
n lt -4
x 4