Expressions & Equations

transcript

www.njctl.org2013-01-23

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Table of Contents

Inverse OperationsOne Step EquationsTwo Step EquationsMulti-Step EquationsDistributing Fractions in Equations

Graphing & Writing Inequalities with One Variable

Click on a topic to go to that section.The Distributive Property and Factoring

Combining Like Terms

Simple Inequalities involving Addition & SubtractionSimple Inequalities involving Multiplication & Division

Common Core Standards: 7.EE.1, 7.EE.3, 7.EE.4

Simplifying Algebraic Expressions

Translating Between Words and Equations

Commutative and Associative Properties

Using Numerical and Algebraic Expressions and Equations

Commutative and Associative Properties

Return to table of contents

Commutative Property of Addition: The order in which the terms of a sum are added does not change the sum.a + b = b + a5 + 7 = 7 + 5 12= 12Commutative Property of Multiplication: The order in which the terms of a product are multiplied does not change the product.ab = ba4(5) = 5(4)

Associative Property of Addition: The order in which the terms of a sum are grouped does not change the sum.(a + b) + c = a + (b + c)(2 + 3) + 4 = 2 + (3 + 4) 5 + 4 = 2 + 7

The Commutative Property is particularly useful when you are combining integers.

Example:-15 + 9 + (-4)=-15 + (-4) + 9= Changing it this way allows for the -19 + 9 = negatives to be added together first. -10

Associative Property of Multiplication: The order in which the terms of a product are grouped does not change the product.

1 Identify the property of -5 + 3 = 3 + (-5)A Commutative Property of Addition B Commutative Property of Multiplication C Associative Property of Addition D Associative Property of Multiplication

2 Identify the property of a + (b + c) = (a + c) + bA Commutative Property of Addition B Commutative Property of Multiplication C Associative Property of Addition D Associative Property of Multiplication

3 Identify the property of (3 * (-4)) * 8 = 3 * ((-4) * 8)A Commutative Property of AdditionB Commutative Property of MultiplicationC Associative Property of AdditionD Asociative Property of Multiplication

Discuss why using the Commutative Property would be useful with the following problems:

1. 4 + 3 + (-4)

2. -9 x 3 x 0

3. -5 x 7 x -2

4. -8 + 1 + (-6)

Combining Like Terms

An Algebraic Expression - contains numbers, variables and at least one operation.

Like terms: terms in an expression that have the same variable raised to the same power

Examples:

LIKE TERMS NOT LIKE TERMS

6x and 2x 6x2 and 2x

5y and 8y 5x and 8y

4x2 and 7x2 4x2y and 7xy2

4 Identify all of the terms like 2x

A 5xB 3x2

C 5yD 12yE 2

5 Identify all of the terms like 8y

A 9yB 4y2

C 7yD 8E -18x

6 Identify all of the terms like 8xy

A 8xB 3x2yC 39xyD 4yE -8xy

7 Identify all of the terms like 2y

A 51wB 2xC 3yD 2wE -10y

8 Identify all of the terms like 14x2

A -5xB 8x2

C 13y2

D xE -x2

If two or more like terms are being added or subtracted, they can be combined.

To combine like terms add/subtract the coefficient but leave the variable alone.

7x +8x =15x9v-2v = 7v

Sometimes there are constant terms that can be combined.9 + 2f + 6 =9 + 2f + 6 =

2f + 15Sometimes there will be both coeffients and constants to

be combined.3g + 7 + 8g - 2

11g + 5Notice that the sign before a given term goes with the

number.

Try These:

1.) 2b +6g(3) + 4f + 9f

2.) 9j + 3 + 24h + 6 + 7h + 3

3.) 7a + 4 + 2a -1 9 + 8c -12 + 5c

4.) 8x + 56xy + 5y

9 8x + 3x = 11xA TrueB False

10 7x + 7y = 14xy

A TrueB False

11 2x + 3x = 5xA TrueB False

12 9x + 5y = 14xyA TrueB False

13 6x + 2x = 8x2

A TrueB False

14 -15y + 7y = -8yA TrueB False

15 -6 + y + 8 = 2yA TrueB False

16 -7y + 9y = 2yA TrueB False

17 9x + 4 + 2x =A 15x B 11x + 4 C 13x + 2x D 9x + 6x

18 12x + 3x + 7 - 5 A 15x + 7 - 5 B 13x C 17x D 15x + 2

19 -4x - 6 + 2x - 14A -22x B -2x - 20 C -6x +20 D 22x

The Distributive Propertyand Factoring

An Area ModelImagine that you have two rooms next to each other. Both are 4 feet long. One is 7 feet wide and the other is 3 feet wide .

How could you express the area of those two rooms together?

Either way, the area is 40 feet2:

You could add 7 + 3 and then multiply by 4

4(7+3)=4(10)=40

You could multiply 4 by 7, then 4 by 3 and add them

4(7) + 4(3) =28 + 12 =40

An Area ModelImagine that you have two rooms next to each other. Both are 4 yards long. One is 3 yards wide and you don't know how wide the other is.

How could you express the area of those two rooms together?

You cannot add x and 3 because they aren't like terms, so you can only do it by multiplying 4 by x and 4 by 3 and adding

4(x) + 4(3)=4x + 12

The area of the two rooms is 4x + 12(Note: 4x cannot be combined with 12)

The Distributive PropertyFinding the area of the rectangles demonstrates the distributive property. Use the distributive property when expressions are written like so: a(b + c)

4(x + 2)4(x) + 4(2)4x + 8

The 4 is distributed to each term of the sum (x + 2)

Write an expression equivalent to:

5(y + 4)5(y) + 5(4)5y + 20

6(x + 2) 3(x + 4)

4(x - 5) 7(x - 1)

Remember to distribute the 5 to the y and the 4

The Distributive Property is often used to eliminate the parentheses in expressions like 4(x + 2). This makes it possible to combine like terms in more complicated expressions.

EXAMPLE:-2(x + 3) = -2(x) + -2(3) = -2x + -6 or -2x - 6

3(4x - 6) = 3(4x) - 3(6) = 12x - 18

-2 (x - 3) = -2(x) - (-2)(3) = -2x + 6TRY THESE:3(4x + 2) =

-1(6m + 4) =

-3(2x - 5) =

Be careful with your signs!

Keep in mind that when there is a negative sign on the outside of the parenthesis it really is a -1.

For example:-(2x + 7) = -1(2x + 7) = -1(2x) + -1(7) = -2x - 7

What do you notice about the original problem and its answer?

The numbers are turned to their opposites.Remove to see answer.

Try these:-(9x + 3) =

-(-5x + 1) =

-(2x - 4) =

-(-x - 6) =

20 4(2 + 5) = 4(2) + 5A TrueB False

21 8(x + 9) = 8(x) + 8(9)

A TrueB False

22 -4(x + 6) = -4 + 4(6)A TrueB False

23 3(x - 4) = 3(x) - 3(4)A TrueB False

24 Use the distributive property to rewrite the expression without parentheses 3(x + 4)

A 3x + 4B 3x + 12C x + 12D 7x

25 Use the distributive property to rewrite the expression without parentheses 5(x + 7)

A x + 35B 5x + 7C 5x + 35D 40x

26 Use the distributive property to rewrite the expression without parentheses (x + 5)2

A 2x + 5B 2x + 10C x + 10D 12x

27 Use the distributive property to rewrite the expression without parentheses 3(x - 4)

A 3x - 4B x - 12C 3x - 12D 9x

28 Use the distributive property to rewrite the expression without parentheses 2(w - 6)

A 2w - 6B w - 12C 2w - 12D 10w

29 Use the distributive property to rewrite the expression without parentheses -4(x - 9)

A -4x - 36B x - 36C 4x - 36D -4x + 36

30 Use the distributive property to rewrite the expression without parentheses 5.2(x - 9.3)

A -5.2x - 48.36B 5.2x - 48.36C -5.2x + 48.36D -48.36x

31 Use the distributive property to rewrite the expression without parentheses

We can also use the Distributive Property in reverse. This is called Factoring. When we factor an expression, we find all numbers or variables that divide into all of the parts of an expression.Example:7x + 35 Both the 7x and 35 are divisible by 77(x + 5) By removing the 7 we have factored the problemWe can check our work by using the distributive property to see that the two expressions are equal.

We can factor with numbers, variables, or both.2x + 4y = 2(x + 2y)9b + 3 = 3(3b + 1)-5j - 10k + 25m = -5(j + 2k - 5m) *Careful of your signs4a + 6a + 8ab = 2a(2 + 3 + 4b)

Try these:Factor the following expressions:1.) 6b + 9c = 2.) -2h - 10j = 3.) 4a + 20ab + 12abc =

32 Factor the following: 4p + 24qA 4 (p + 24q)B 2 (2p + 12q)C 4(p + 6q)D 2 (2p + 24q)

33 Factor the following: 5g + 15h

A 3(g + 5h)B 5(g + 3h)C 5(g + 15h)D 5g (1 + 3h)

34 Factor the following: 3r + 9rt + 15rxA 3(r+ 3rt + 5rx)B 3r(1 + 3t + 5x)C 3r (3t + 5x)D 3 (r + 9rt + 15rx)

35 Factor the following: 2v+7v+14vA 7(2v + v + 2v)B 7v(2 + 1 + 2)C 7v (1 + 2)D v(2 + 7 + 14)

36 Factor the following: -6a - 15ab - 18abc

A -3a(2 + 5b + 6bc)B 3a(2+ 5b + 6bc) C -3(2a - 5b - 6bc)D -3a (2 -5b - 6bc)

- What divides into the expression: -5n - 20mn - 10np

- If a regular pentagon has a perimeter of 10x + 25, what does each side equal?

Simplifying Algebraic Expressions

Now we will use what we know about combining like terms and the distributive property to simplify

algebraic expressions.

Remember, like terms have the same variable and same exponent.

To simplify:

4 + 5(x + 3) First Distribute

4 + 5(x) + 5(3)

4 + 5x + 15 Then combine Like Terms

5x + 19

Notice that when combining like terms, you add/subtract the coefficients but the variable remains the same. Remember that you can combine coefficient or constant terms.

37 7x +3(x - 4) = 10x - 4A TrueB False

38 8 +(x + 3)5 = 5x + 11 A TrueB False

39 4 +(x - 3)6 = 6x -14A TrueB False

40 2x + 3y + 5x + 12 = 10xy + 12A TrueB False

41 5x2 + 2x + 7(x + 1) + x2 = 6x2 + 9x + 7A TrueB False

42 2x3 + 4x2 + 6(x2 + 3x) + x = 2x3 + 10x2 + 4x A TrueB False

43 The lengths of the sides of home plate in a baseball field are represented by the expressions in the accompanying figure.

A 5xyzB x2 + y3zC 2x + 3yzD 2x + 2y + yz

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Which expression represents the perimeter of the figure?

44 A rectangle has a width of x and a length that is double that. What is the perimeter of the rectangle?

A 4xB 6xC 8xD 10x

Inverse Operations

What is an equation?

An equation is a mathematical statement containing an equal sign to show that two expressions are equal.

2 + 3 = 5

9 – 2 = 7

5 + 3 = 1 + 7

An algebraic equation is just an equation that has algebraic symbols in one or both of the expressions.

4x = 24

9 + h = 15

Equations can also be used to state the equality of two expressions containing one or more variables.

In real numbers we can say, for example, that for any given value of x it is true that

4x + 1 = 13

x = 3 because

4(3) + 1 = 13 12 + 1 = 13 13 = 13

An equation can be compared to a balanced scale.

Both sides need to contain the same quantity in order for it to be "balanced".

For example, 9+ 11 = 6 + 14 represents an equation because both sides simplify to 20. 9 + 11 = 6 + 14

20 = 20

Any of the numerical values in the equation can be represented by a variable.

Examples:

15 + c = 25

x + 10 = 25

15 + 10 = y

When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).

In order to solve an equation containing a variable, you need to use inverse (opposite/undoing) operations on both sides of the equation.

Let's review the inverses of each operation:

Addition Subtraction

Multiplication Division

Square Square Root

There are two questions to ask when solving an equation:*What operation is in the equation?*What is the inverse of that operation (This will be the operation you use to solve the equation.)?

A good phrase to remember when doing equations is:Whatever you do to one side of the equation, you do to the other.For example, if you add three on one side of the equal sign you must add three to the other side as well…to keep the equation in balance.

To solve for "x" in the following equation... x + 7 = 32

Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides (in this case, it is subtraction).

x + 7 = 32 - 7 -7 x = 25

In the original equation, replace x with 25 and see if it makes the equation true.

x + 7 = 3225 + 7 = 32 32 = 32

For each equation, write the inverse operation needed to solve for the variable.

a.) y +7 = 14 subtract 7 b.) a - 21 = 10 add 21

c.) 5s = 25 divide by 5 d.) x = 5 multiply by 12

Think about this...

To solve c - 3 = 12

Which method is better? Why?

Kendra

Added 3 to each side of the equation

c - 3 = 12 +3 +3 c = 15

Subtracted 12 from each side, then added 15.

c - 3 = 12 -12 -12c - 15 = 0 +15 +15 c = 15

45 What is the inverse operation needed to solve this equation?

2x = 14A AdditionB SubtractionC MultiplicationD Division

46 What is the inverse operation needed to solve this equation?

x - 3 = -12

A AdditionB SubtractionC MultiplicationD Division

47 What is the inverse operation needed to solve this problem? -2 + x = 9

A Addition B Subtraction C Multiplication D Division

One Step Equations

To solve equations, you must work backwards through the order of operations to find the value of the variable.

Remember to use inverse operations in order to isolate the variable on one side of the equation.

Whatever you do to one side of an equation, you MUST do to the other side!

Examples:

y + 3 = 13 - 3 -3 The inverse of adding 3 is subtracting 3 y = 10

4m = 32 4 4 The inverse of multiplying by 4 is dividing by 4 m = 8

Remember - whatever you do to one side of an equation, you MUST do to the other!!!

x - 5 = 2 +5 +5 x = 7

x + 5 = -14 -5 -5 x = -19

2 = x - 4+4 +4 6 = x

6 = x + 1-1 -1 5 = x

12 = x + 17-17 -17 -5 = x

x + 9 = 5 -9 -9 x = -4

One Step EquationsSolve each equation then click the box to see work &

solution.

click to showinverse operation

One Step Equations 4x = 16 4 4 x = 4

-2x = -12 -2 -2 x = 6

-20 = 5x 5 5 -4 = x

x 2x = 18

= 9 (2) (2)

x-6 x = -216

= 36 (-6)(-6)

click to showinverse operation

48 Solve.

x - 7 = 19

49 Solve.

j + 15 = 17

50 Solve.

42 = 6y

51 Solve.

-115 = -5x

52 Solve.

= 12 x 9

53 Solve.

w - 17 = 37

54 Solve.

-3 = x 7

55 Solve.

23 + t = 11

56 Solve.

108 = 12r

Sometimes the operation can be confusing.For example: -2 + x = 7This looks like you should use subtraction to undo the problem. However, -2 + x = 7 is the same as x - 2 = 7 so while it appears to be addition, it is really subtraction. In order to undo this we can add.-2 + x = 7x - 2 = 7 +2 +2 x = 9

OR OR -2 + x = 7 -2 + x = 7- (-2) -(-2) +2 +2

x = 9 x = 9

-2 + x = 7-2 = -2-4 + x = 5This did not cancel out anything.

-2 + x = 7+2 +2 x = 9This did cancel out to find the answer.

-2 + x = 7 x - 2 = 7 +2 +2 x = 9This is the same as the middle problem

Try these:

1.) -4 + b = 7

2.) -2 + r = 4

3.) -3 + w = 6

4.) -5 + c = 9

Think about this...

In the expression

To which does the "-" belong?

Does it belong to the x? The 3? Both?

The answer is that there is one negative so it is used once with either the variable or the 3. Generally, we assign it to the 3 to avoid creating a negative variable.

57 Solve.

58 Solve.

-5 + q = 15

59 Solve.

60 Solve

61 Solve.

62 Solve.

63 Solve.

Sometimes you will have an equation where you are multiplying a variable by a fraction.

𝟑𝟒 𝒙=𝟗

To undo the fraction you:Multiply by the Reciprocal of the Coefficent

This means that you will flip the fraction and then multiply

**Dividing by a fraction is the same as multiplying by its reciprocal

1 times any number is itself so this is why it can cancel out.

64 Solve.

65 Solve

66 Solve.

Two-Step Equations

Sometimes it takes more than one step to solve an equation. Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable.

This means that you undo in the opposite order (PEMDAS):

1st: Addition & Subtraction2nd: Multiplication & Division3rd: Exponents4th: Parentheses

Whatever you do to one side of an equation, you MUST do to the other side!

Examples:

4x + 2 = 10 - 2 - 2 Undo addition first 4x = 8 4 4 Undo multiplication second x = 2

-2y - 9 = -13 + 9 + 9 Undo subtraction first -2y = -4 -2 -2 Undo multiplication second

Remember - whatever you do to one side of an equation, you MUST do to the other!!!

5b + 3 = 18 -3 -3 5b = 15 5 5 b = 3

3j - 4 = 14 +4 +4 3j = 18 3 3 j = 6

-2x + 3 = -1 - 3 -3 -2x = -4 -2 -2 x = 2

Two Step EquationsSolve each equation then click the box to see

work & solution.

-2m - 4 = 22 +4 +4 -2m = 26 -2 -2 m = -13

+5 = +5

t = 15

w + 6 = 102 -6 -6

w 2 = 4 2 2 w = 8

67 Solve the equation.

5x - 6 = -56

14 = 3c + 2

- 4 = 24

5r - 2 = -12

14 = -2n - 6

+ 7 = 13 x 5

+ 2 = -10 x 3-

78 Solve

1 2x + = 1

Multi-Step Equations

Steps for Solving Multiple Step Equations

As equations become more complex, you should:

1. Simplify each side of the equation.(Combining like terms and the distributive property)

2. Use inverse operations to solve the equation.

Remember, whatever you do to one side of an equation, you MUST do to the other side!

Examples:

5x + 7x + 4 = 28 12x + 4 = 28 Combine Like Terms -4 - 4 Undo Addition 12x = 24

12 12 Undo Multiplication x = 2

-1 = 2r - 7r +19 -1 = -5r + 19 Combine Like Terms-19 = - 19 Undo Subtraction-20 = -5r -5 -5 Undo Multiplication 4 = r

Try these.12h - 10h + 7 = 25

-17q + 7q -13 = 27

17 - 9f + 6 = 140

q = - 4

f = -13

Always check to see that both sides of the equation are simplified before you begin solving the equation.

Sometimes, you need to use the distributive property in order to simplify part of the equation.

Remember: The distributive property is a(b + c) = ab + ac

Examples5(20 + 6) = 5(20) + 5(6) 9(30 - 2) = 9(30) - 9(2)

3(5 + 2x) = 3(5) + 3(2x)

-2(4x - 7) = -2(4x) - (-2)(7)

Examples:

2(b - 8) = 28 2b - 16 = 28 Distribute the 2 through (b - 8) +16 +16 Undo subtraction 2b = 44 2 2 Undo multiplication b = 22

3r + 4(r - 2) = 13 3r + 4r - 8 = 13 Distribute the 4 through (r - 2) 7r - 8 = 13 Combine Like Terms +8 +8 Undo subtraction 7r = 21 7 7 Undo multiplication

Try these.

3(w - 2) = 9

4(2d + 5) = 92

6m + 2(2m + 7) = 54

81 Solve.

9 + 3x + x = 25

82 Solve

-8e + 7 +3e = -13

83 Solve.

-27 = 8x - 4 - 2x - 11

84 Solve.

n - 2 + 4n - 5 = 13

85 Solve.

32 = f - 3f + 6f

86 Solve.6g - 15g + 8 - 19 = -38

87 Solve.

3(a - 5) = -21

88 Solve.

4(x + 3) = 20

89 Solve.

3 = 7(k - 2) + 17

90 Solve.

2(p + 7) -7 = 5

91 Solve.

3m -1m + 3(m-2) = 19.75

92 Solve.

93 Solve.

94 Solve.

Distributing Fractions in Equations

Remember...

1. Simplify each side of the equation.

2. Solve the equation.(Undo addition and subtraction first, multiplication and division second)

Remember, whatever you do to one side of an equation, you MUST do to the other side!

There is more than one way to solve an equation with a fraction coefficient. While you can, you don't need to distribute.

Multiply by the reciprocal Multiply by the LCD

(-3 + 3x) = 3 5

-3 + 3x = 24+3 +3 3x = 27 3 3 x = 9

(-3 + 3x) = 3 5

3(-3 + 3x) = 72 -9 + 9x = 72 +9 +9 9x = 81 9 9 x = 9

Some problems work better when you multiply by the reciprocal and some work better multiplying by the LCM.

Which strategy would you use for the following? Why?

95 Solve.

96 Solve.

(8 - 3c) = 2 3

97 Solve.

98 Solve.

99 Solve.

Translating Between Words and Expressions

List words that indicate addition

List words that indicate subtraction

List words that indicate multiplication

List words that indicate division

List words that indicate equals

Be aware of the difference between "less" and "less than".

For example:"Eight less three" and "Three less than Eight" are

equivalent expressions. So what is the difference in wording?

Eight less three: 8 - 3Three less than eight: 8 - 3

When you see "less than", you need to switch the order of the numbers.

As a rule of thumb, if you see the words "than" or "from" it means you have to reverse the order

of the two items on either side of the word.

Examples: ·8 less than b means b - 8·3 more than x means x + 3·x less than 2 means 2 - x

_______________

_______________click to reveal

_______________

The many ways to represent multiplication...

How do you represent "three times a"?

(3)(a) 3(a) 3 a 3a

The preferred representation is 3a

When a variable is being multiplied by a number, the number (coefficient) is always written in front of the variable.

The following are not allowed:3xa ... The multiplication sign looks like another variablea3 ... The number is always written in front of the variable

Representation of division...

How do you represent "b divided by 12"?

b ÷ 12

b ∕ 12

When choosing a variable, there are some that are often avoided:

l, i, t, o, O, s, S

Why might these be avoided?

It is best to avoid using letters that might be confused for numbers or operations. In the case above (1, +, 0, 5)

Three times j

Eight divided by j

j less than 7

5 more than j

4 less than j

TRANSLATE THE WORDS INTO AN

ALGEBRAIC EXPRESSION

÷÷÷

23 + m

The sum of twenty-three and m

Write the expression for each statement.Then check your answer.

d - 24

Twenty-four less than d

4(8-j)

Write the expression for each statement.Remember, sometimes you need to use parentheses for a quantity.

Four times the difference of eight and j

The product of seven and w, divided by 12

(6+p)2

The square of the sum of six and p

100 The quotient of 200 and the quantity of p times 7

A 200 7p

B 200 - (7p)

C 200 ÷ 7p

D 7p 200

101 35 multiplied by the quantity r less 45

A 35r - 45B 35(45) - rC 35(45 - r)D 35(r - 45)

102 Mary had 5 jellybeans for each of 4 friends.

A 5 + 4 B 5 - 4 C 5 x 4 D 5 ÷ 4

103 If n + 4 represents an odd integer, the next larger odd integer is represented by

A n + 2B n + 3C n + 5D n + 6

104 a less than 27

A 27 - aB a

27C a - 27D 27 + a

105 If h represents a number, which equation is a correct translation of:“Sixty more than 9 times a number is 375”?

A 9h = 375B 9h + 60 = 375C 9h - 60 = 375D 60h + 9 = 375

Using Numerical and Algebraic Expressions and

Equations

We can use our algebraic translating skills to solve other problems.

We can use a variable to show an unknown.A constant will be any fixed amount.

If there are two separate unknowns, relate one to the other.

The school cafeteria sold 225 chicken meals today. They sold twice the number of grilled chicken sandwiches than chicken tenders. How many of each were sold?

2c + c = 225

chickensandwiches chicken

tenderstotal meals

c + 2c = 225 3c = 225 3 3 c = 75

The cafeteria sold 150 grilled chicken sandwiches and 75 tenders.

Julie is matting a picture in a frame. Her frame is 9 inches wide and her picture is 7 inches wide. How much matting should she put on either side?

2m + 7 = 92m + 7 = -7 -7 2m = 2 2 2 m = 1Julie needs 1 inches on each side.

both sides of the mat size of

picturesize of frame

Many times with equations there will be one number that will be the same no matter what (constant) and one that can be changed based on the problem (variable and coefficient).

Example: George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $127.00. How many games did he buy in all?

George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $127.00. How many games did he buy in all?

Notice that the video games are "per game" so that means there could be many different amounts of games and therefore many different prices. This is shown by writing the amount for one game next to a variable to indicate any number of games.

cost ofone videogame

number of games

Notice also that there is a specific amount that is charged no matter what, the flat fee. This will not change so it is the constant and it will be added (or subtracted) from the other part of the problem.

30g + 7

number of games

the cost of the shipping

"Total" means equal so here is how to write the rest of the equation.

30g + 7 = 127

numberof games

the total amount

the cost of the shipping

Now we can solve it.

30g + 7 = 127

-7 -7 30g = 120 30 30

g = 4 George bought 4 video games.

106 Lorena has a garden and wants to put a gate to her fence directly in the middle of one side. The whole length of the fence is 24 feet. If the gate is 4 feet, how many feet should be on either side of the fence?

107 Lewis wants to go to the amusement park with his family. The cost is $12.00 for parking plus $27.00 per person to enter the park. Lewis and his family spent $147. Which equation shows this problem?

A 12p + 27 = 147 B 12p + 27p = 147 C 27p + 12 = 147 D 39p = 147

108 Lewis wants to go to the amusement park with his family. The cost is $12.00 for parking plus $27.00 per person to enter the park. Lewis and his family spent $147. How many people went to the amusement park WITH Lewis?

109 Mary is saving up for a new bicycle that is $239. She has $68.00 already saved. If she wants to put away $9.00 per week, how many weeks will it take to save enough for her bicycle? Which equation represents the situation?

A 9 + 68 = 239B 9d + 68 = 239C 68d + 9 = 239D 77d = 239

110 Mary is saving up for a new bicycle that is $239. She has $68.00 already saved. If she wants to put away $9.00 per week, how many weeks will it take to save enough for her bicycle?

111 You are selling t-shirts for $15 each as a fundraiser. You sold 17 less today then you did yesterday. Altogether you have raised $675.

Write and solve an equation to determine the number of t-shirts you sold today.

Be prepared to show your equation!

112 Rachel bought $12.53 worth of school supplies. She still needs to buy pens which are $2.49 per pack. She has a total of $20.00 to spend on school supplies. How many packs of pens can she buy?

Write and solve an equation to determine the number of packs of pens Rachel can buy.

113 The length of a rectangle is 9 cm greater than its width and its perimeter is 82 cm.

Write and solve an equation to determine the width of the rectangle.

114 The product of -4 and the sum of 7 more than a number is -96.

Write and solve an equation to determine the number. Be prepared to show your equation!

115 A magazine company has 2,100 more subscribers this year than last year. Their magazine sells for $182 per year. Their combined income from last year and this year is $2,566,200.

Write and solve an equation to determine the number of subscribers they had each year.

How many subscribers last year?

116 A magazine company has 2,100 more subscribers this year than last year. Their magazine sells for $182 per year. Their combined income from last year and this year is $2,566,200.

Write and solve an equation to determine the number of subscribers they had each year.

How many subscribers this year?

117 The perimeter of a hexagon is 13.2 cm.

Write and solve an equation to determine the length of a side of the hexagon.

Graphing and WritingInequalities

with One Variable

When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.

Important Words

Sample Sentence

Equivalent Translation

is more than Trenton is more than 10 miles away.

t > 10

is greater than A is greater than B.

must exceed The speed must exceed 25 mph.

The speed is greater than 25 mph.

s > 25

When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.

Important Words

Sample Sentence

Equivalent Translation

cannot exceed Time cannot exceed 60 minutes.

Time must be less than or equal to 60 minutes.

t < 60

is at most At most, 7 students were late for class.

Seven or fewer students were late for class.

is at least Bob is at least 14 years old.

Bob's age is greater than or equal to 14.

B > 14

How are these inequalities read?

2 + 2 > 3 Two plus two is greater than 3

2 + 2 ≥ 4 Two plus two is greater than or equal to 4

2 + 2 < 5 Two plus two is less than 5

2 + 2 ≤ 5 Two plus two is less than or equal to 5

2 + 2 ≤ 4 Two plus two is less than or equal to 4

2 + 2 > 3 Two plus two is greater than or equal to 3

Writing inequalities

Let's translate each statement into an inequality.

x is less than 10

20 is greater than or equal to y

x < 10

inequality statement

translate to

20 > y

You try a few:

1. 14 is greater than a

2. b is less than or equal to 8

3. 6 is less than the product of f and 20

4. The sum of t and 9 is greater than or equal to 36

5. 7 more than w is less than or equal to 10

6. 19 decreased by p is greater than or equal to 2

7. Fewer than 12 items

8. No more than 50 students

9. At least 275 people attended the play

Do you speak math?Change the following expressions from English into math.

Double a number is at most four.

Three plus a number is at least six.

2x ≤ 4

3 + x ≥ 6

Answer

Five less than a number is less than twice that number.

The sum of two consecutive numbers is at least thirteen.

Three times a number plus seven is at least nine.

x - 5 < 2x

x + (x + 1) ≥ 13

3x + 7 > 9

Answer

0 1 2 3 4 5 6 7 8 9 10

at least

An employee earns

A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour.

Let e represent an employee's wages.

Try this:

The speed limit on a road is 55 miles per hour. Define a variable and write an inequality.

118 You have $200 to spend on clothes. You already spent $140 and shirts cost $12. Which equation shows this scenario?

A 200 < 12x + 140B 200 12x + 140C 200 > 12x + 140D 200 12x + 140

119 A sea turtle can live up to 125 years. If one is already 37 years old, which scenario shows how many more years could it live?

125 < 37 + x125 37 + x≤

ABC 125 > 37 + x D 125 37 + x≥

120 The width of a rectangle is 3 in longer than the length. The perimeter is no less than 25 inches.

A 4a + 6 < 25B 4a + 6 25C 4a + 6 > 25D 4a + 6 ≥ 25

121 The absolute value of the sum of two numbers is less than or equal to the sum of the absolute values of the same two numbers.

A solution to an inequality is NOT a single number. It will have more than one value.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

This would be read as the solution set is all numbers greater than or equal to negative 5.

Solution Sets

Let's name the numbers that are solutions of the given inequality.

r > 10 Which of the following are solutions? {5, 10, 15, 20}

5 > 10 is not trueSo, not a solution

10 > 10 is not trueSo, not a solution

15 > 10 is trueSo, 15 is a solution

20 > 10 is trueSo, 20 is a solution

Answer:{15, 20} are solutions of the inequality r > 10

Let's try another one.30 ≥ 4d; {3, 4, 5, 6, 7, 8}

30 ≥ 4d30 ≥ (4)330 ≥ 12

30 ≥ 4d30 ≥ (4)430 ≥ 16

30 ≥ 4d30 ≥ (4)530 ≥ 20

30 ≥ 4d30 ≥ (4) 630 ≥ 24

30 ≥ 4d30 ≥ (4)730 ≥ 28

30 ≥ 4d30 ≥ (4)830 ≥ 32

click to reveal click to reveal click to reveal

click to revealclick to revealclick to reveal

Graphing Inequalities - The CircleAn open circle on a number shows that the number is not part of the solution. It is used with "greater than" and "less than". The word equal is not included.< >

A closed circle on a number shows that the number is part of the solution. It is used with "greater than or equal to" and "less than or equal to". < >

Graphing Inequalities - The Arrow

The arrow should always point in the direction of those numbers who satisfy the inequality.

*If the variable is on the left side of the inequality, then < and ≤ will show an arrow pointing left.

*If the variable is on the left side of the inequality, then > and ≥ will show an arrow pointing right.

Notice that < and ≤ look like an arrow pointing left and that > and ≥ look like an arrow pointing right.

But what if the variable isn't on the left? Do the opposite of where the inequality symbol points.

-1 0-2-3-4-5 1 2 3 4 5

What is the number in the inequality?

What kind of circle should be used?

In what direction does the line go?

Graphing Inequalities

Step 1: Rewrite this as x < 5.

Step 2: What kind of circle? Because it is less than, it does not include the number 5 and so it is an open circle.

-1 0-2-3-4-5 1 2 3 4 5

Graphing Inequalitiesx is less than 5

Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality.

Step 3: Draw an arrow on the number line showing all possible solutions. Numbers greater than the variable, go to the right. Numbers less than the variable, go to the left.

-1 0-2-3-4-5 1 2 3 4 5

Step 1: Rewrite this as x ≤ 5.

Step 2: What kind of circle? Because it is less than or equal to, it does include the number 5 and so it is a closed circle.

-1 0-2-3-4-5 1 2 3 4 5

Graphing Inequalitiesx is less than or equal to 5

Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality.

Step 3: Draw an arrow on the number line showing all possible solutions. Numbers greater than the variable, go to the right. Numbers less than the variable, go to the left.

-1 0-2-3-4-5 1 2 3 4 5

x ≤ 5

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

You tryGraph the inequalityx > 2

Graph the inequality -3 > x

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

click 2 on the number line for answer

click -3 on the number line for answer

.05.05

Try these.Graph the inequalities.1. x > -3

-1 0-2-3-4-5 1 2 3 4 5

2. x < 4

-1 0-2-3-4-5 1 2 3 4 5

Try these.State the inequality shown.1.

-1 0-2-3-4-5 1 2 3 4 5

122 This solution set would be x > -4.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

A TrueB False

-1 0-2-3-4-5 1 2 3 4 5

A x > 3

B x < 3

C x < 3

D x > 3

State the inequality shown.

5 6 7 8 9 10 11 12 13 14 15

A 11 < xB 11 > xC 11 > xD 11 < x

-1 0-2-3-4-5 1 2 3 4 5

A x > -1B x < -1C x < -1D x > -1

-1-5 1 50-2-3-4 2 3 4

A -4 < xB -4 > xC -4 < xD -4 > x

-1 0-2-3-4-5 1 2 3 4 5

A x > 0B x < 0C x < 0D x > 0

Simple Inequalities Involving Additionand Subtraction

x + 3 = 13 - 3 - 3 x = 10

Remembers how to solve an algebraic equation?

Does 10 + 3 = 13 13 = 13Be sure to check your answer!

Use the inverse of addition

· Solving one-step inequalities is much like solving one-step equations. ·To solve an inequality, you need to isolate the variable using the properties of inequalities and inverse operations.

· Remember, whatever you do to one side, you do to the other.

12 > x + 5-5 -5 Subtract to undo addition 7 > x

To find the solution, isolate the variable x.

Remember, it is isolated when it appears by itself on one side of the equation.

7 > xThe symbol is > so it is an open circle and it is numbers less than 7 so it goes to the left.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

A. j + 7 > -2

Solve and graph.

-9 is not included in solution set; therefore we graph with an open circle.

A. j + 7 > -2 -7 -7 j > -9

B. r - 2 > 4

Solve and graph.

1110 12 13 149876543210

r - 2 > 4 +2 +2 r > 6

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

5 > w- 4 - 4

9 > w + 4

C. 9 > w + 4

Solve and graph.

128 Solve the inequality.

3 < s + 4

____ < s

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10A

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10B

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10C

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10D

129 Solve the inequality and graph the solution.-4 + b < -2

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10A

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10B

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10C

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10D

130 Solve the inequality and graph the solution.-8 > b - 5

131 Solve the inequality.

m + 6.4 < 9.6m < ______

Simple Inequalities Involving Multiplication

and Division

Since x is multiplied by 3, divide both sides by 3 for the inverse operation.

Multiplying or Dividing by a Positive Number

3x > -36

3x > -36 3 3

x > -12

Solve the inequality.

3 2( )

Since r is multiplied by 2/3, multiply both sides by the reciprocal of 2/3.

r < 4 3 2( )

132 3k > 18

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10A

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10D

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2

-3-4-5-6-7-8-9-10

-30 > 3q

10 > q-10 < q-10 > q

D 10 < q

134 X 2

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10D

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

g > 27

g > 36

g > 108g > 36

D g > 108

-21 > 3d

d > -7d > -7d < -7

D d < -7

·Sometimes you must multiply or divide to isolate the variable.

·Multiplying or dividing both sides of an inequality by a negative number gives a surprising result.

Now let's see what happens when we multiply or divide by negative numbers.

1. Write down two numbers and put the appropriate inequality (< or >) between them.

2. Apply each rule to your original two numbers from step 1 and simplify. Write the correct inequality (< or >) between the answers.

A. Add 4

B. Subtract 4

C. Multiply by 4

D. Multiply by -5

E. Divide by 4

F. Divide by -4

3. What happened with the inequality symbol in your results?

4. Compare your results with the rest of the class.

5. What pattern(s) do you notice in the inequalities?

How do different operations affect inequalities?

Write a rule for inequalities.

Let's see what happens when we multiply this inequality by -1.

5 > -1

-1 • 5 ? -1 • -1 -5 ? 1

-5 < 1

We know 5 is greater than -1

Multiply both sides by -1

Is -5 less than or greater than 1?

You know -5 is less than 1, so you should use <

What happened to the inequality symbol to keep the inequality statement true?

The direction of the inequality changes only if the number you are using to multiply or divide by is negative.

Helpful Hint

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Dividing each side by -3 changes the > to <.

-3y > 18

-3y < 18 -3 -3

y < -6

Solve and graph.

Divide each side by -7

Flip the sign because you divided by a negative.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

-7m > -28

-7m < -28 -7 -7

Solve and graph.

Divide each side by 5.

The sign does NOT change because you did not divide by a negative.

5m > -255m > -25 5 5

m > -5

Solve and graph.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

D. -8y > 32

Solve and graph.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10E. -9f > -54

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

You multiplied by a negative.

-r 2 < 5

-2( )r > -10

Multiply both sides by the reciprocal of -1/2.

-r 2 > 5 -2( )

Why did the inequality change?

1. -6h < 42

Try these.Solve and graph each inequality.

2. 4x > -2010 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

3. 5m < 30

Try these.Solve and graph each inequality.

4. > -3 a -2 10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve and graph.

3y < -6

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve and graph.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve and graph.-5y ≤ -25

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve and graph.

Expressions & Equations

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