7th grade math expressions & equations -...

expressionsandequations7thgrade20120705.notebook

1

October 26, 2012

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New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative

www.njctl.org

7th Grade Math

Expressions & Equations

www.njctl.org

20120705

table of contents

Table of Contents

Inverse OperationsOne Step EquationsTwo Step EquationsMultiStep EquationsVariables on Both SidesMore EquationsGraphing & Writing Inequalities with One Variable

Click on a topic to go to that section.

The Distributive PropertyCombining Like Terms

Simple Inequalities involving Addition & Subtraction

Simple Inequalities involving Multiplication & Division

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

Apr 258:15 PM

The Distributive Property

Return to Table of Contents

Apr 258:19 PM

An Area ModelFind the area of a rectangle whose width is 4 and whose length is x + 2

4

x 2

Area of two rectangles: 4(x) + 4(2) = 4x + 8

4

x + 2

Area of One Rectangle:4(x+2) = 4x + 8

Apr 258:41 PM


Finding the area of the rectangles demonstrates the distributive property

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

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


2

October 26, 2012

Apr 258:41 PM

Write an expression equivalent to:

5(x + 3)5(x) + 5(3)5x + 15

6(x + 4) 5(x + 7)

2(x 1) 4(x 8)

Apr 258:41 PM

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

3(4x 6) = 3(4x) 3(6) = 12x 18 or 12x + 18

TRY THESE:6(2x + 4) =

1(5m 8) =

(x + 5) =

(3x 6) =

Apr 258:50 PM

1 4(2 + 5) = 4(2) + 5

True

False

Apr 258:51 PM

2 8(x + 9) = 8(x) + 8(9)

True False

Apr 258:52 PM

3 4(x + 6) = 4 + 4(6)

True False

Apr 258:53 PM

4 3(x 4) = 3(x) 3(4)

True False


3

October 26, 2012

Apr 258:53 PM


a(b + c) = ab + ac Example: 2(x + 3) = 2x + 6

(b + c)a = ba + ca Example: (x + 7)3 = 3x + 21

a(b c) = ab ac Example: 5(x 2) = 5x 10

(b c)a = ba ca Example: (x 3)6 = 6x 18click to reveal

Apr 259:02 PM

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

A 2x + 5B 2x + 10

C x + 10D 7x

Apr 259:02 PM

6 Use the distributive property to rewrite the expression without parentheses 3(x + 7)

A x + 21

B 3x + 7

C 3x + 21

D 24x

Apr 259:02 PM

7 Use the distributive property to rewrite the expression without parentheses (x + 6)3

A 3x + 6

B 3x + 18

C x + 18

D 21x

Apr 259:02 PM

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

A 3x 4

B x 12

C 3x 12

D 9x

Apr 259:02 PM

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

A 2w 6

B w 12

C 2w 12

D 10w


4

October 26, 2012

Apr 259:02 PM

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

A 4x 36

B x 36

C 4x 36

D 32x

Apr 259:02 PM

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

A 5.2x 48.36

B 5.2x 48.36

C 5.2x + 48.36

D 48.36x

Apr 259:02 PM

12 Use the distributive property to rewrite the expression without parentheses

A

B

C

D

Mar 2910:29 AM

Combining Like Terms


Oct 1212:30 PM

Expression contains numbers, variables and at least one operation.

Apr 259:11 PM

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

Examples:

LIKE TERMS NOT LIKE TERMS6x and 2x 6x2 and 2x

5y and 8y 5x and 8y

4x2 and 7x2 4x2y and 7xy2


5

October 26, 2012

Apr 259:17 PM

13 Identify all of the terms like 5y

A 5B 4y2

C 18y

D 8y

E 1y

Apr 259:17 PM

14 Identify all of the terms like 8x

A 5x

B 4x2

C 8y

D 8E 10x

Apr 259:17 PM

15 Identify all of the terms like 8xy

A 5x

B 4x2y

C 3xy

D 8y

E 10xy

Apr 259:17 PM

16 Identify all of the terms like 2y

A 51y

B 2w

C 3y

D 2xE 10y

Apr 259:17 PM

17 Identify all of the terms like 14x2

A 5x

B 2x2

C 3y2

D 2xE 10x2

Apr 259:26 PM

Simplify by combining like terms

6x + 3x9x

5x + 2x 7x

4 + 5(x + 3)4 + 5(x) + 5(3)4 + 5x + 15 5x + 19

7y 4y 3y

Notice that when combining like terms, you add/subtract the coefficients but the variable remains the same.


6

October 26, 2012

Apr 259:26 PM

Try These:

8x + 9x

7y + 5y

6 + 2x + 12x

7y + 7x

Apr 259:44 PM

18 8x + 3x = 11x

True False

Apr 259:44 PM

19 7x + 7y = 14xy

True False

Apr 259:45 PM

20 4x + 4x = 8x2

True False

Apr 259:45 PM

21 12y + 4y = 8y

True False

Apr 259:45 PM

22 3 + y + 5 = 2y

True False


7

October 26, 2012

Apr 259:45 PM

23 3y + 5y = 2y

True False

Apr 259:45 PM

24 7x +3(x 4) = 10x 4

True False

Apr 259:45 PM

25 7 +(x + 2)5 = 5x + 9

True False

Apr 259:45 PM

26 4 +(x 3)6 = 6x 14

True False

Apr 259:45 PM

27 3x + 2y + 4x + 12 = 9xy + 12

True False

Apr 259:45 PM

28 3x2 + 7x + 5(x + 3) + x2 = 4x2 + 12x + 15

True False


8

October 26, 2012

Apr 259:45 PM

29 9x3 + 2x2 + 3(x2 + x) + 5x = 9x3 + 5x2 + 6x

True False

Jul 287:32 PM

30 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

yz

yy

xx

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?

variables

Inverse Operations


equation

What is an equation?

An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in

2+3=5

92=7

equation

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 = 14 1

If x = 3, then

4(3) + 1 = 14 1 12 + 1 = 13

13 = 13

equation

When defining your variables, remember...

Letters from the beginning of the alphabet like a, b, c... often denote constants in the context of the discussion at hand.

While letters from end of the alphabet, like x, y, z..., are usually reserved for the variables, a convention initiated by Descartes.

Try It!

Write an equation with a variable and have a classmate identify the variable and its value.


9

October 26, 2012

Apr 118:22 PM

An equation can be compared to a balanced scale.

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

inverse

For example, 20 + 30 = 50 represents an equation because both sides simplify to 50. 20 + 30 = 50

50 = 50

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

Examples:

20 + c = 50 x + 30 = 50

20 + 30 = y

Aug 3110:50 PM

Why are we Solving Equations?

First we evaluated expressions where we were given the value of the variable and had to find what the expression simplified to.

Now, we are told what it simplifies to and we need to find the value of the variable.

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).

inverse

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

Oct 309:46 PM

There are four properties of equality that we will use to solve equations. They are as follows:

Addition PropertyIf a=b, then a+c=b+c for all real numbers a, b, and c. The same number can be added to each side of the equation without changing the solution of the equation.

Subtraction PropertyIf a=b, then ac=bc for all real numbers a, b, and c. The same number can be subtracted from each side of the equation without changing the solution of the equation.

Multiplication PropertyIf a=b, and c=0, then ac=bc for all real numbers ab, b, and c. Each side of an equation can be multiplied by the same nonzero number without changing the solution of the equation.

Division PropertyIf a=b, and c=0, then a/c=b/c for all real numbers ab, b, and c. Each side of an equation can be divided by the same nonzero number without changing the solution of the equation.

inverse

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.

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


10

October 26, 2012

inverse

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 12

move move

move move

inverse

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

Ted

Subtracted 12 from each side, then added 15.

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

inverse

Think about this...

In the expression

To which does the "" belong?

Does it belong to the x? The 5? Both?

The answer is that there is one negative so it is used once with either the variable or the 5. Generally, we assign it to the 5 to avoid creating a negative variable.

So:

question

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

7x = 49

A Addition

B Subtraction

C Multiplication

D Division

question

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

x 3 = 12

A Addition

B Subtraction

C Multiplication

D Division

OneStep Equations

One Step Equations



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October 26, 2012

Apr 1111:18 PM

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!

Apr 1111:19 PM

Examples:

y + 9 = 16 9 9 The inverse of adding 9 is subtracting 9 y = 7

6m = 72 6 6 The inverse of multiplying by 6 is dividing by 6 m = 12

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

one step add/ subtract

x 8 = 2 +8 +8 x = 6

x + 2 = 14 2 2 x = 16

2 = x 6+6 +6 8 = x

7 = x + 33 3 4 = x

15 = x + 1717 17 2 = x

x + 5 = 3 5 5 x = 2

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

click to showinverse operation






one step mult/divide

One Step Equations

3x = 15 3 3 x = 5

4x = 12 4 4 x = 3

25 = 5x 5 5 5 = x




x 2x = 20

= 10 (2) (2)

x6 x = 216

= 36


(6)(6)


question

33 Solve.

x 6 = 11

question

34 Solve.

j + 15 = 17


12

October 26, 2012

question

35 Solve.

115 = 5x

question

36 Solve.

= 12 x 9

question

37 Solve.

51 = 17y

question

38 Solve.

w 17 = 37

question

39 Solve.

3 = x 7

question

40 Solve.

23 + t = 11


13

October 26, 2012

question

41 Solve.

108 = 12r

question

42 Solve.

question

43 Solve.

question

44 Solve.

question

45 Solve.

question

46 Solve.


14

October 26, 2012

TwoStep Equations

TwoStep Equations


Apr 1111:18 PM

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!

Apr 1111:19 PM

Examples:

3x + 4 = 10 4 4 Undo addition first 3x = 6 3 3 Undo multiplication second

x = 2

4y 11 = 23 + 11 +11 Undo subtraction first 4y = 12 4 4 Undo multiplication second

y = 3

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

two step practice

67x = 836 6 7x = 77 7 7 x = 11

3x + 10 = 46 10 10 3x = 36 3 3 x = 12

4x 3 = 25 +3 +3 4x = 28 4 4 x = 7

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

9 + 2x = 239 9 2x = 14 2 2 x = 7

8 2x = 88 8 2x = 16 2 2 x = 8

Two Step Equations

Solve each equation then click the box to see work & solution.

Apr 1211:25 PM

47 Solve the equation.

5x 6 = 56

Apr 1211:25 PM


16 = 3m 8


15

October 26, 2012

Apr 1211:25 PM


x 2

6 = 30

Apr 1211:25 PM


5r 2 = 12

Apr 1211:25 PM


12 = 2n 4

Apr 1211:25 PM


7 = 13 x 4

Apr 1211:25 PM


+ 3 = 12 x 5

Apr 1211:25 PM



16

October 26, 2012

Apr 1211:25 PM


Apr 1211:25 PM


Apr 1211:25 PM


Apr 1211:25 PM


Combining Like Terms

MultiStep Equations


Apr 1111:18 PM

Steps for Solving Multiple Step EquationsAs 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!


17

October 26, 2012

Apr 1111:19 PM

Examples:

15 = 2x 9 + 4x15 = 2x 9 Combine Like Terms +9 +9 Undo Subtraction first 6 = 2x 2 2 Undo Multiplication second 3 = x

7x 3x 8 = 244x 8 = 24 Combine Like Terms + 8 +8 Undo Subtraction first4x = 32 4 4 Undo Multiplication second x = 8

practice

Now try an example. Each term is infinitely cloned so you can pull them down as you solve.

7x + 3 + 6x = 6

answer

practice

Now try another example. Each term is infinitely cloned so you can pull them down as you solve.

6x 5 + x = 44

answer

info table

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.

info table

For all real numbers a, b, c

a(b + c) = ab + ac

a(b c) = ab ac

Distributive Property

info table

Examples

5(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)


18

October 26, 2012

Apr 1111:19 PM

Examples:

5(1 + 6x) = 185 5 + 30x = 185 Distribute the 5 on the left side 5 5 Undo addition first 30x = 180 30 30 Undo multiplication second x = 6

2x + 6(x 3) = 142x + 6x 18 = 14 Distribute the 6 through (x 3)8x 18 = 14 Combine Like Terms +18 +18 Undo subtraction 8x = 32

8 8 Undo multiplication x = 4

equation ex

5 ( 2 + 7x ) = 95

Now show the distributing and solve...(each number/ symbol is infinitely cloned, so click on it and drag another one down)

equation ex

6 ( 2x + 9 ) = 102

Now show the distributing and solve...(each number/ symbol is infinitely cloned, so click on it and drag another one down)

Apr 1211:49 PM

59 Solve.

3 + 2t + 4t = 63

Apr 1211:49 PM

60 Solve.

19 = 1 + 4 x

Apr 1211:49 PM

61 Solve.

8x 4 2x 11 = 27


19

October 26, 2012

Apr 1211:49 PM

62 Solve.

4 = 27y + 7 (15y) + 13

Apr 1211:49 PM

63 Solve.

9 4y + 16 + 11y = 4

Apr 1211:49 PM

64 Solve.

6(8 + 3b) = 78

Apr 1211:49 PM

65 Solve.

18 = 6(1 1k)

Apr 1211:49 PM

66 Solve.

2w + 8(w + 3) = 34

Apr 1211:49 PM

67 Solve.

4 = 4x 2(x + 6)


20

October 26, 2012

Apr 1211:49 PM

68 Solve.

3r r + 2(r + 4) = 24

Apr 1211:49 PM

69 Solve.

Apr 1211:49 PM

70 Solve.

Apr 1211:49 PM

71 Solve.

Apr 1211:49 PM

72 Solve.

Apr 1211:49 PM

73 Solve.


21

October 26, 2012

Variables on Both Sides

Variables on Both Sides


Apr 1111:18 PM

Remember...

1. Simplify both sides of the equation.

2. Collect the variable terms on one side of the equation. (Add or subtract one of the terms from both sides of the equation)

3. Solve the equation.


Apr 1111:19 PM

Example:

4x + 8 = 2x + 262x 2x Subtract 2x from both sides2x + 8 = 26 8 8 Undo Addition 2x = 18 2 2 Undo Multiplication

x = 9

What if you did it a little differently?4x + 8 = 2x + 264x 4x Subtract 4x from both sides 8 = 2x + 26 26 26 Undo Addition 18 = 2x 2 2 Undo Multiplication

9 = x

Recommendation: Cancel the smaller amount of the variable!

Apr 1111:19 PM

Example:

6r 5 = 7r + 7 2r 6r 5 = 5r + 7 Simplify Each Side of Equation5r 5r Subtract 5r from both sides (smaller than 6r) r 5 = 7 + 5 +5 Undo Subtraction r = 12

practice

Try these:

6x 2 = x + 13 4(x + 1) = 2x 2 5t 8 = 9t 10x x 4x + 4 = 2x 2 5t 5t 5x 2 = 13 2x 2x 8 = 4t 10 + 2 +2 2x + 4 = 2 +10 +105x = 15 4 4 2 = 4t 5 5 2x = 6 4 4 x = 3 2 2 = t

x = 3

1 2

what if/

Sometimes, you get an interesting answer.What do you think about this?What is the value of x?

3x 1 = 3x + 1

Since the equation is false, there is "no solution"!

No value will make this equation true.move this


22

October 26, 2012

what if/

Sometimes, you get an interesting answer.What do you think about this?What is the value of x?

3(x 1) = 3x 3

Since the equation is true, there are infinitely many solutions! The equation is called an identity.

Any value will make this equation true.

move this

practice

Try these:

4y = 2(y + 1) + 3(y 1) 14 (2x + 5) = 2x + 9 9m 8 = 9m + 4 4y = 2y + 2 + 3y 3 14 2x 5 = 2x + 9 9m 9m 4y = 5y 1 9 2x = 2x + 9 8 = 45y 5y +2x +2x No Solution y = 1 9 = 9 y = 1 Identity

example

Mary's distance (rate time) equals Jocelyn's distance

(rate time)

Mary and Jocelyn left school at 3:00 p.m. and bicycled home along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?

Define t = Mary's time in hourst + 0.25 = Jocelyn's time in hours

Relate

Write 12t = 9(t+0.25)

example

12t = 9(t + 0.25)

12t = 9t + 2.259t 9t

3t = 2.253 3

t = 0.75

It took Mary 0.75h, or 45 min, to get home.

Step 1 distribute the 9 inside the parenthesis(pull)

Step 2 subtract 9t from both sides(pull)

Step 3 divide both sides by 3(pull)

Apr 1211:49 PM

74 Solve.

7f + 7 = 3f + 39

Apr 1211:49 PM

75 Solve.

h 4 = 5h + 26


23

October 26, 2012

Apr 1211:49 PM

76 Solve.

w 2 + 3w = 6 + 5w

Apr 1211:49 PM

77 Solve.

5(x 5) = 5x + 19

Apr 1211:49 PM

78 Solve.

4m + 8 2(m + 3) = 4m 8

Apr 1211:49 PM

79 Solve.

28 7r = 7(4 r)

Rational Numbers and Equations

More Equations


rational numbers

Remember...

1. Simplify each side of the equation.

2. Collect the variable terms on one side of the equation. (Add or subtract one of the terms from both sides of the equation)

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



24

October 26, 2012

Apr 1111:19 PM

Examples:

x = 6

x = 6 Multiply both sides by the reciprocal of

x =

x = 10

2x 3 = + xx x Subtract x from both sides x 3 =

+3 +3 Undo Subtractionx =

3 5 3 5

5 3

5 3

5 3

30 3

14 5

14 5

15

Apr 1111:19 PM

There is more than one way to solve an equation with distribution.

Multiply by the reciprocal Multiply by the LCM

(3 + 3x) = 3 5

72 5

(3 + 3x) = 3 5

72 5

(3 + 3x) = 3 5

72 5

(3 + 3x) = 3 5

72 5

5 3

5 3

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

(3 + 3x) = 3 5

72 5

5 5

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

Apr 146:13 PM

80 Solve

3 5

1 2

x + = 1 10

Apr 146:13 PM

81 Solve

1 5

2b + 5b = 68 35

Apr 146:13 PM

x + 8 = 7 + x

82 Solve

2 3

Apr 146:13 PM

(8 3c) =

83 Solve

2 3

16 3


25

October 26, 2012

Apr 146:13 PM

6(7 3y) + 4y = 10(2y 4)

84 Solve

Apr 146:13 PM

(6 2z) = z (4z + 6)

85 Solve

1 2

9 4

1 8

Apr 146:13 PM

9.47x = 7.45x 8.81

86 Solve

Apr 146:13 PM

13.19 8.54x = 7.94x 1.82

87 Solve

Apr 146:13 PM

3(8 2m) + 8m = 4(4 + m)

88 Solve

Apr 146:13 PM

(2y 4) = 3(y + 2) 3y

89 Solve

1 2


26

October 26, 2012

Aug 1012:59 PM

90 You are selling tshirts 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 tshirts you sold today.

Be prepared to show me your equation!

Aug 1012:59 PM

91 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.


Aug 1012:59 PM

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

Write and solve an equation to determine the number.


Aug 1012:59 PM

93 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.


Aug 1012:59 PM

94 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/Writing

Graphing and WritingInequalities

with One Variable



27

October 26, 2012

Oct 510:56 AM

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

is greater than

must exceed

Oct 510:56 AM

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

is at most

is at least

Reading inequalities

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

Writing inequalitiesLet's translate each statement into an inequality.

x is less than 10

20 is greater than or equal to y

x < 10

words

inequality statement

translate to

20 > y

Writing Inequalities

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

Answ

ers


Do you speak math?Try to change the following expressions from English into math.

Twice a number is at most six.

Two plus a number is at least four.

2x ≤ 6

2 + x ≥ 4

Answer

Answer


28

October 26, 2012


Three less than a number is less than three times that number.

The sum of two consecutive numbers is at least thirteen.

Three times a number plus one is at least ten.

x 3 < 3x

x + (x + 1) ≥ 13

3x + 1 > 10

Answer

Answer

Answer

Solution Sets

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 1012345678910

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

Solution Sets

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

Solution Sets

Let's try another one.

30 ≥ 5d; 4,5,6,7,8

30 ≥ 5d30 ≥ 5(4)30 ≥ 20

30 ≥ 5d30 ≥ 5(5)30 ≥ 25

30 ≥ 5d30 ≥ 5(6)30 ≥ 30

30 ≥ 5d30 ≥ 5(7)30 ≥ 35

30 ≥ 5d30 ≥ 5(8)30 ≥ 40

Answer: 4,5,6

Graphing Inequalities

Graphing Inequalities with Greater/Less Than or Equal To

An 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".< >


Remember!

Open circle means that number is not included in the solution set and is used to represent < or >.

Closed circle means the solution set includes that number and is used to represent ≤ or ≥.


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October 26, 2012


Do you know where to start?

How do represent the starting point?

Is there a special symbol?



Step 1: Figure out what the inequality solution requires. For example, rewrite x is less than one as x < 1.

Step 2: Draw a circle on the number line where the number being graphed is represented. In this case, an open circle since it represents the starting point for the inequality solution but is not part of the solution.

1 02345 1 2 3 4 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. For numbers greater than the variable, shade to the right of the boundary point. For numbers less than the variable, shade to the left of the boundary point.

1 02345 1 2 3 4 5

1 02345 1 2 3 4 5

x < 1


10 2 3 4 5 6 7 8 9 1012345678910

Step 1: Figure out what the inequality solution requires. For example, rewrite x is greater than or equal to one as x > 1.

Step 2: Draw a circle on the number line where the number being graphed is represented. In this case, a closed circle since it represents the starting point for the inequality solution and is a part of the solution.



10 2 3 4 5 6 7 8 9 1012345678910

You try

Graph the inequalityx > 5

Graph the inequality 3 > x

10 2 3 4 5 6 7 8 9 1012345678910


Try these.Graph the inequalities.

1. x > 4

1 02345 1 2 3 4 5

2. x < 5

1 02345 1 2 3 4 5


30

October 26, 2012


Try these.State the inequality shown.

1.

1 02345 1 2 3 4 5

1 02345 1 2 3 4 5

2.


95 Would this solution set be x > 4?

True

False

10 2 3 4 5 6 7 8 9 1012345678910

Apr 137:21 PM

Remember!

Closed circle means the solution set includes that number and is used to represent ≤ or ≥.

Open circle means that number is not included in the solution set and is used to represent < or >.


1 02345 1 2 3 4 596

A x > 3

B x < 3

C x < 3

D x > 3


5 6 7 8 9 10 11 12 13 14 15

97

A 11 < x

B 11 > x

C 11 > x

D 11 < x


1 02345 1 2 3 4 5

98

A x > 1

B x < 1

C x < 1

D x > 1


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October 26, 2012


15 1 599

A 4 < x

0234 2 3 4

B 4 > x

C 4 < x

D 4 > x


100

A x > 0

1 02345 1 2 3 4 5

B x < 0

C x < 0

D x > 0

Graphing Inequlities

0 1 2 3 4 5 6 7 8 9 10

7.5

$7.50

7.5

at least

>

An employee earns

e

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. Graph the solutions.

Let e represent an employee's wages.

Graphing Inequlities

Try this:

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

Answ

er

Simple Inequalities

Simple Inequalities Involving Additionand Subtraction


Simple Inequalites

x + 3 = 13 3 3 x = 10

Who remembers how to solve an algebraic equation?

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

Use the inverse of addition


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October 26, 2012

Apr 138:05 PM

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

Simple Inequalities

12 > x + 6

To find the solution, isolate the variable x.

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

Simple Inequalities

Step 1: Since 6 is added to x and subtraction is the inverse of addition, subtract 6 from both sides to undo the addition.

12 > x + 6 6 6

6 > x

Simple Inequalities

Step 2: Check the computation. Substitute the end point of 6 for x. The end point is not included (open circle) since x < 6.

12 > x + 612 > 6 + 612 > 12

0 1 2 3 4 5 6 7 8 9 10

Simple Inequalities

Step 3: Check the direction of the inequality. Choose a number from your line (such as 4) and check that it fits the inequality.

0 1 2 3 4 5 6 7 8 9 10

6 > x6 > 4

Simple Inequalities

10 2 3 4 5 6 7 8 9 1012345678910

k > 5

3 3k + 3 > 2

A. k + 3 > 2

Solve and graph.

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


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October 26, 2012

Simple Inequalities

r > 11

+ 9 +9r 9 > 2

B. r 9 > 2

Solve and graph.

1110 12 13 149876543210

Simple Inequalities

10 2 3 4 5 6 7 8 9 1012345678910

5 > w

4 49 > w + 4

w < 5

C. 9 > w + 4

Solve and graph.

Simple Inequalities Add/Sub

1 02345 1 2 3 4 5

5 6

2A

1 02345 1 2 3 4 5B

1 02345 1 2 3 4 5C

1 02345 1 2 3 4 5D

1 2

1 3

101 Solve the inequality and graph the solution.

n 2 >

2

2

2

5 6

5 6

5 6


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A

10 2 3 4 5 6 7 8 9 1012345678910

B

10 2 3 4 5 6 7 8 9 1012345678910

C

10 2 3 4 5 6 7 8 9 1012345678910

D


2 < s + 8


10 2 3 4 5 6 7 8 9 1012345678910

A

10 2 3 4 5 6 7 8 9 1012345678910

B

10 2 3 4 5 6 7 8 9 1012345678910

C

10 2 3 4 5 6 7 8 9 1012345678910

D


6 + b < 4


10 2 3 4 5 6 7 8 9 1012345678910

A

10 2 3 4 5 6 7 8 9 1012345678910

B

10 2 3 4 5 6 7 8 9 1012345678910

C

10 2 3 4 5 6 7 8 9 1012345678910

D


5 > b 2


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October 26, 2012


10 2 3 4 5 6 7 8 9 1012345678910

1.5

1.5

A

10 2 3 4 5 6 7 8 9 1012345678910

B

10 2 3 4 5 6 7 8 9 1012345678910

1.5C

10 2 3 4 5 6 7 8 9 1012345678910

D


3.5 < m + 2

1.5

Simple Inequalities Mult/Div

Simple Inequalities Involving Multiplication

and Division



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

10 2 3 4 5 6 7 8 9 1012345678910

Multiplying or Dividing by a Positive Number

3x > 27

3x > 27 3 3

x > 9

Remove for Graph


Solve the inequality and graph the solution.

2 3

r < 6

3 2( )

r < 9

10 2 3 4 5 6 7 8 9 1012345678910

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

2 3

r < 6 3 2( )

Remove for Graph


106 4k > 24

10 2 3 4 5 6 7 8 9 1012345678910

A

B

C

10 2 3 4 5 6 7 8 9 1012345678910D

10 2 3 4 5 6 7 8 9 1012345678910

10 2 3 4 5 6 7 8 9 1012345678910


107

A

B

C

50 > 5q

10 > q

10 < q

10 > q

D 10 < q


35

October 26, 2012


108 X 2

A

B

C

D

10 2 3 4 5 6 7 8 9 1012345678910

10 2 3 4 5 6 7 8 9 1012345678910

10 2 3 4 5 6 7 8 9 1012345678910

10 2 3 4 5 6 7 8 9 1012345678910

< 1


109

A

B

C

g > 27

g > 36

g > 108

g > 36

D g > 108

3 4


110

A

B

C

28 > 4d

d > 7

d > 7

d < 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

Apr 1410:19 PM

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.


36

October 26, 2012


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?


Words OriginalInequality

Multiply/Divide by aNegative #

Result

Multiplying or dividing by a negative number reverses the inequality symbol

3 > 1 Multiply by2 6 < 2

4 < 12 Divide by 4 1 > 3

Apr 151:34 PM

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 1012345678910

Dividing each side by 3 changes the > to <.

3y > 15

3y < 15 3 3

y < 5

Solve and graph.

A.


Divide each side by 7

10 2 3 4 5 6 7 8 9 1012345678910

7m < 21

7m < 21 7 7

m < 3

Solve and graph.

B.


Divide each side by 5.

5m > 25

5m > 25 5 5

m > 5

Solve and graph.

C.

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37

October 26, 2012


D. 8y > 24

Solve and graph.

10 2 3 4 5 6 7 8 9 1012345678910

10 2 3 4 5 6 7 8 9 1012345678910

E. 9f > 45


10 2 3 4 5 6 7 8 9 1012345678910

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. 7h < 49

Try these.Solve and graph each inequality.

2. 3x > 15

10 2 3 4 5 6 7 8 9 1012345678910

10 2 3 4 5 6 7 8 9 1012345678910


3. 7m < 21

Try these.Solve and graph each inequality.

4. > 2

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10 2 3 4 5 6 7 8 9 1012345678910

a 2


111

10 2 3 4 5 6 7 8 9 1012345678910

Solve and graph.

2y < 4

A y < 2

B y > 2

C y > 2

D y > 2


112

10 2 3 4 5 6 7 8 9 1012345678910

Solve and graph.

x 1< 4

A x < 4

B x > 4

C x < 3

D x > 3


38

October 26, 2012


113

10 2 3 4 5 6 7 8 9 1012345678910

Solve and graph.

5y ≤ 25

A y < 5

B y > 5

C y < 5

D y > 5


114

10 2 3 4 5 6 7 8 9 1012345678910

Solve and graph.

n 2> 3

A n < 6

B n > 6

C n < 6

D n > 6

Apr 156:41 PM

REMEMBER:

An inequality stays the same when you:

1. Add, subtract, multiply or divide by the same positive number on both sides

2. Add or subtract the same negative number on both sides

An inequality changes direction when you:

1. Multiply or divide by the same negative number on both sides

7th grade math expressions & equations -...

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