Unit 1 Algebra Basics

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

In this example, we have add first and do the exponent.

(3)2 9

Example 2: 3 ⨯-4 ÷ 2 -12 ÷ 2

In this example, we have multiplication and division. We always work from left to right.

-6

Example 3: (42 ÷ 2) - 12 (16 ÷ 2) – 12 8 – 12 -4

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

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.

-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.

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!