Unit 1 Algebra Basics

Review

Written and Compiled by Genny Simpson

Lesson 1

Integer Rules

To add integers:

If the integers have the same sign, add them

and keep the sign. For example: -3 + -6 =-9

and 7 + 12 = 19.

If the integers have different signs, ignore the

signs, subtract them and take the sign of the

larger number. For example: -10 + +15 = +5

and -16 + 4 = -12.

Lesson 1

Integer Rules

To subtract integers:

Use the definition of subtraction which is

adding the opposite. For example if you have

4 – 8, that is the same thing as 4 + -8. Now

you are right back to addition! You ignore the

signs, subtract and take the sign of the larger

number: 4 + -8 = -4.

Let’s take a look at another example: -5 - -14.

If we use the definition of subtraction, we

would have -5 + 14. Two negative signs, side

by side, “bump each other out” and make a

plus. We are now back to our addition rules:

if we have opposite signs, we ignore the

signs, subtract and take the sign of the larger

number, so -5 + 14 = +9.

Lesson 1

Integer Rules

To multiply and/or divide integers:

The rules are simple for multiplying and

dividing integers. If you have an even number

of negative signs, then the answer is positive.

For example, -3 ⨯-4 = +12. We have two

negative signs. Two is an even number, so

our answer is positive. This is also true for

division. -36 ÷-4 = +9.

What if we have different signs or an odd

number of negative signs? For example:

-12 ⨯ 3 = -36 or 32 ÷ -4 = -8. In each of these

examples, there is one negative sign, so our

answers are negative.

Integer Rules

Now it’s your turn! Try these problems on a

sheet of paper. Make sure you write down the

problem and show your work.

1. -12 + 9 2. 15 – 22

3. -13 – 18 4. -5 + -6

5. 11 – -12 6. -12 - -18

7. -17 ⨯ -2 8. 4 ⨯ -5

9. -27 ÷ -3 10. 35 ÷ -7

Lesson 2

Order of Operations

When we simplify expressions it is important that

we follow the same order. Without a certain

order, everyone could have a different answer!

Remember PEMDAS!

P is for parentheses or any other type of

grouping symbol.

E is for exponents.

M is for multiplication from left to right.

D is for division from left to right.

A is for addition from left to right.

S is for subtraction from left to right.

Examples of Using Order of Operations to

Simplify Expressions

Example 1: (-2 + 5)2

(3)2

9

Example 2: 3 ⨯-4 ÷ 2

-12 ÷ 2

-6

Example 3: (42 ÷ 2) - 12

(16 ÷ 2) – 12

8 – 12

-4

In this example, we have add first

and do the exponent.

In this example, we have

multiplication and division. We

always work from left to right.

In this example, we have an

exponent inside the parentheses.

We have to do the exponent first.

Order of Operations

Now it’s your turn! Make sure you write

down the problem, show your work, and be

neat!

1. 6 – (2 ⨯ -3) 2. 32 + 12 ÷ -2

3. -12 + 23 – 8 4. (4 + -8)2 ⨯ -3

5. 4 - 5 ⨯ 2 6. 24 ÷ -3 + 7 ⨯ -2

7. -6 – 6 ⨯ -1 8. 4(4 – 6) + 3

9. 25 ÷ -5 ⨯-3 – 6 10. (23 + -6)2 ⨯ -3*

Think about #10. Work from the inside to the outside.

Lesson 3

Evaluating Variable Expressions

There are three things to remember when

you are evaluating variable expressions:

1. Write down the problem.

2. Substitute in a value for each variable.

3. Simplify using order of operations.

Let’s look at some examples.

Example 1: ab2 – 2a; if a = -3 and b = 4

-3(4)2 – 2(-3)

-3(16) – (-6)

-48 + 6

-42

In this example, we needed PEMDAS and our

integer rules. Remember that a number in

front of parentheses means multiplication.

Example 2: cd – d; c = 3 and d = 7

3(7) – 7

21 – 7

14

Example 3: (p2 – q) ÷ (p + q); p = 4 and q = -9

(16 - -9) ÷ (4 + -9)

(16 + 9) ÷ (-5)

25 ÷ (-5)

-5

Now don’t get stressed out. Just remember to

substitute and simplify.

Evaluating Variable Expressions

It’s your turn again! Remember to write

down the problem, substitute the numbers in

for the variables, and simplify using order of

operations and the integer rules.

1. n2 – m; n = 7 and n = 8

2. a + ac; a = 15 and c = 8

3. q ÷ 6 + p; p = 10 and q = 12

4. 7(c – d); c = 3 and d = 9

5. ab ÷ 3; a = 12 and b = -4

6. 15 – (m – p); m = 4 and p = 12

7. 10 − x + y ÷ 2; x = 7 and y = -18

8. p2 ÷ 4 − m; m = 3, and p = 4

9. zy + 4y; y = 5, and z = 2

10. mn ÷ 6 + 10; m = 7, and n = 6

Lesson 4

Combining Like Terms and the Distributive Property

What exactly are like terms? They are terms that

have the same variable or variables and the same

exponent.

Here are some examples of like terms.

2a and -6a; 12cd and 20cd; -5p2 and 10p2

As long as terms are alike or the same, you

can combine them by adding or subtracting

the numbers that are in front of them. Those

numbers in front of the variables are called

coefficients.

For example: 2a + -6a = (2 + -6)a = -4a

12cd – 20cd = (12 – 20)cd = -8cd

-5p2 + 10p2 = (-5 + 10)p2 = 5p2

In each of these examples, we simply added or

subtracted the numbers in front of the variables.

Now let’s take a look at some examples of using

the distributive property. The important thing to

remember here is to distribute all the way

through the parentheses.

Example 1: 3(a + 5) = 3 ⦁ a + 3 ⦁ 5 = 3a + 15

Example 2: (2b – 7)-4 = 2b ⦁ -4 + -7 ⦁ -4 = -8b + 28

In the next two examples, let’s use the

distributive property along with combining like

terms.

Example 3: 5(2c + 9) – 7c

5 ⦁ 2c + 5 ⦁ 9 – 7c

10c + 45 – 7c

10c – 7c + 45

(10 – 7)c + 45

3c + 45

In this example I used the

distributive property first. I then

put the like terms together so

that I could combine them. I

cannot combine 3c and 45

because they are not like terms.

Example 4: 5a – 4(2a + 12)

5a + -4(2a + 12)

5a + -4 ⦁ 2a + -4 ⦁ 12

5a + -8a + -48

(5 + -8)a + -48

-3a + -48

-3a – 48

When you are using the distributive property

along with combining like terms, remember to

distribute first, rearrange things to put like terms

together and combine only those terms that look

exactly alike except for the numbers in front.

Don’t worry if you make a mistake, you can

always erase and start over.

In this example I used the

definition of subtraction first.

I then used the distributive

property. Next, I combined

like terms. I cannot combine

-3a and -48 because they are

not like terms.

Combining Like Terms and the Distributive Property

Okay, it’s your turn one more time! Remember to use

the definition of subtraction: subtraction is adding the

opposite, distribute, rearrange terms to put like terms

together, and combine only those terms that are

exactly alike.

1. -3(2w – 5) 2. (12 + 5a)4

3. 5(3 + 2h) 4. -2(4 – 3c)

5. (6b – 7)(-3) 6. 4(2m + 5) – 11m

7. 5k – 10(2k + 9) 8. -5(3ab + 2) + 6ab

9. 12pq + 4(3 – 4pq) 10. 3(2z + 3) – 4(2z + 9)*

*Number 10 looks hard, but remember to use the definition of

subtraction, distribute, rearrange to put like terms together, and

combine like terms. You can do it!