An equation is a mathematical sentence that contains an = sign.

555 = A-75 A-75 = 555 are examples of an equation.

In the examples above “A” is a variable which represents an unknown number.

One way to solve for A is to complete the Inverse (opposite) Operation (the opposite of subtraction is addition)

555 + 75 = A or A = 555 + 75, so A = 630

1. T - 5 = 11

11 + 5 = T

16 = 16

3. B + 6 = 15

B = 15 - 6

B = 9

2. 16 - T = 11

T = 16 - 11

T = 5

4. 9 + B = 15

B = 15 - 9

B = 6

Find the example that does not use the inverse Operation.

Memorize this type of problem so you can solve using the same operation

5. 6 x R = 48

48 6 = R

8 = R

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6. W x 8 = 48

48 8 = w

6 = w

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7. 72 S = 9

72 9 = S

8 = S

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8. M 9 = 8

8 x 9 = M

72 = M

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Find the example that does not use the inverse Operation.

Memorize this type of problem so you can solve using the same operation

Equations are like a Balance scale. If weight is added or subtracted to one side it will make the scale tip or not be balanced. In order to keep the scale balanced you must add or subtract equal amounts to both sides.

This idea will help us solve equations with variables.

Property of Equality for Addition and Subtraction

F - 244 =120

F - 244 + 244 = 120 + 244

Add 244 to both sides

F = 364

Solve

Addition Property of Equality If you add the same number to each side of an equation, then the 2 sides remain the same.

X + 3.6 = 12.4

X + 3.6 - 3.6 = 12.4 - 3.6

X = 8.8

Subtract 3.6 from both sides

Solve

Subtraction Property of Equality If you subtract the same number from each side of an equation, then the 2 sides remain the same.

Property of Equality for Multiplication and Division

434 = 2s

434 = 2s 2 2

Divide both sides by 2

S = 217

Solve

Division Property of Equality If each side of an equation is divided by the same nonzero number, then the 2 sides remain equal.

n 8 = 96

n 8 x 8 = 96 x 8

n = 768

Multiply both sides by 8

Solve

Multiplication Property of Equality If each side of an equation is multiplied by the same number, then the 2 sides remain equal.

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Remember a variable next to a number indicates multiplication

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An expression is one part of an equation Examples

15 - 6 20 + a

7 - x 42 d

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÷Numerical expressions are often written in sentence form.

Five more hits than the Yankees Ten fewer points than the Knicks

The key words often indicate what to do

Y + 5 K - 10

Yankees + 5 Knicks - 10

or

more means add

fewer means subtract

Addition

Plus

Sum

More than

Increased by

Total

Subtraction

Minus

Difference

Less than

Subtract

Decreased by

Multiplication

Times

Product

multiplied

Division

Divided

quotient

3x = 18 3x + 9 = 18 How are these equations different?

When one side of an equation has 2 or more operations we need more than one step to solve.

3x + 9 = 18 First subtract 9 from both sides.

3x + 9 - 9 = 18 - 9 Simplify

3x = 9 Then divide each side by 3

3x = 93 3

So x = 3

B - 0.8 = 1.39

First add 0.8 to both sides

B - 0.8 + 0.8 = 1.3 + 0.8 9

Simplify

B = 2.19

Then multiply both sides by 9

B x 9 = 2.1 x 99

So B = 18.9

An integer is a whole number that can be either greater than 0, called positive, or less than 0, called negative. Zero is neither positive nor negative.

Two integers that are the same distance from zero in opposite directions are called opposites.

Every integer on the number line has an absolute value, which is its distance from zero. The brackets indicate the absolute value.

-5 is the opposite of 5

Using a Number Line to Add or Subtract Integers

Add a positive integer by moving to the right on the number line

Add a negative integer by moving to the left on the number line

2 62 + 6 = ?

8 + (- 3 ) = ?8 -3

3 - 7 = ?

3

-7

Subtract an integer by adding its opposite

3 - (-7) = ?

3

7

Another way to remember the rule for subtraction is Keep, change, change!

3 - 7 = ?

Keep the first number

3

Change the sign

+

Change the sign of the second number

-7

4 - 9 = ?

Keep the 4

4

Change the sign

+Change the sign of the 9

(-9)

To multiply or divide signed integers, always multiply or divide the absolute values and use these rules to determine the sign of the answer:

If the signs are the same the product or quotient is Positive

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

If the signs are different the product or quotient is Negative

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

++++

Using Counters to Solve Integer Problems

+ _

A positive and a negative counter equal 0

To add 4 + (-3) we start with 4 positive counters

___ We put 3 negative counters in

The positive and negative pairs cancel each other out to make 0

We are left with 1 positive counter

To add we put counters in the box

Subtracting Integers With Counters

++++

To subtract we take counters out of the box

4 - (-5) Since there are no negative counters we must add negatives to the box. If we add 0 to the box we do not change the value+_

+_

+_

+_

+_

+_

If we take out the 5 negatives we will be left with 9 positives

+

+

+

+

+

+

+

+

+

4 - (-5) = 9

Using Counters to Multiply Integers

In the sentences 3 x 2 we will place 3 groups of 2 in the box

++

++

++

++

++

++

Multiplying a Positive Integer by a Negative Integer

In the problem 3 x (-2) we are putting in 3 groups of 2 negatives in the box

__

__

__

__

__

__

The box shows that 3 x (-2) = - 6

Multiplying With a Negative Integer

When multiplying by a negative integer we are taking out groups

In the problem (-2 ) x 3 we are taking out 2 groups of 3

Since the box is empty we must add 0 pairs until we have enough to take out 2 groups of 3

_+

_+

_+

_+

_+

_+

_+

_+

_+

_+

_+

_+

We can now take out 2 groups of 3

_

_

_

_

_

_

(-2) x 3 = -6

We are left with 6 negatives

3x + 9 = 18 3x + 9 > 18 How are these number sentences different?

When an equation has > or < sin it is called an inequality

3x + 9 > 18 First subtract 9 from both sides.

3x + 9 - 9 > 18 - 9 Simplify

3x > 9 Then divide each side by 3

3x > 93 3

So x > 3

We solve inequalities in the same way as equations

We can graph the solution to an inequality on a number line.

2a - 5 > 9 First, solve as you would an equation

2a - 5 + 5 > 9 + 5 Add 5 to both sides

2a > 92 2

Divide both sides by 2

a > 2.5 Since a is greater than 2.5 it will include all numbers greater than 2.5 but not 2.5

We can draw a circle at 2.5 to show this.

coordinate planeThe plane determined by a horizontal number line, called the x-axis,..

and a vertical number line, called the y-axis,

intersecting at a point called the origin

Each point in the coordinate plane can be specified by an ordered pair of numbers

(-3,1)

Here's one way geometry is used in the real world. A team of archaeologists is studying the ruins of Lignite, a small mining town from the 1800's. They plot points on a coordinate plane to show exactly where each artifact is found.

Name each

Point.

(1,3)

(-5,2) (-2,4)

(3,-4) (5,-6)

(5,-4)