2022 Mid-Term Review

2022 Mid-Term Review

Absolute Value ​

​Absolute value bars look like: | -3 |

​Absolute Value

• ​measures the distance from 0 on a number line

• ​is always POSITIVE

​

​

​

• ​Examples:

Multiple Choice

On a number line, what is the distance between -17 and 9?

• 8

• -8

• -26

• 26

Multiple Choice

Simplify: | 21 | - | -6 |

• 15

• -15

• 27

• -27

Multiple Choice

Simplify: 12 + | 8 | - | 10 |

• 6

• -6

• 10

• 30

Integers​

includes all whole numbers and their opposites (all positive and negative whole numbers)

​Integers Vocabulary for Word Problems

Multiple Choice

Which integer represents an 8-yard loss?

• -8

• 8

Multiple Choice

Which integers represents ascending 510 feet?

• -510

• 510

​Adding and Subtracting Integer Rules

​Adding Integers:

• ​SAME SIGN: add and keep the sign

  • ​Example: 9 + 9 = 18

  • ​Example: -5 + (-2) = -7

• ​DIFFERENT SIGNS: subtract and take the sign of the bigger number

  • ​Example: 9 + (-4) = 5

  • ​Example: -12 + 8 = -4

​Subtracting Integers:

• ​Keep, Change, Change (additive inverse: make it an addition problem)

• ​and then follow rules for adding integers

Multiple Choice

-6 + (-6) =

• 0

• 12

• -12

Multiple Choice

18 + (-5) =

• 13

• -13

• 23

• -23

Multiple Choice

-22 + 12 =

• 34

• -34

• 10

• -10

Multiple Choice

-7 - 4 =

• -3

• 3

• -11

• 11

​Multiplying and Dividing

Integers Rules

• TWO NEGATIVES MAKE A POSITIVE: ​a negative multiplied or divided by a negative is a positive

​

​

• ​A NEGATIVE AND POSITIVE MAKE NEGATIVE: a negative multiplied or divided by positive is a negative

​Multiplying and Dividing Integers Rules

• ​An EVEN number of negatives will give you a POSTIVE answer

  • ​Example: (2)(-5)(-1) = 10

​

• ​An ODD number of negatives will give you a NEGATIVE answer

  • ​Example: (-10)(-2)(-3) = -60

Multiple Choice

-8(-6) =

• 48

• -48

• 56

• -56

Multiple Choice

72 / -24 =

• -3

• 3

• -1728

• 1728

Multiple Choice

(-1)(-1)(-1)(-1)=

• 0

• -4

• 1

• -1

Multiple Choice

(-2)(-24)(3)=

• -144

• 144

• -48

• 48

Adding & Subtracting Fractions & Mixed Numbers with LIKE Denominators​

​Steps:

1. ​Convert mixed numbers to improper fractions if needed

2. ​Add or subtract the numerators

3. ​Leave the denominator the same

4. ​Simplify

​

​

​Step 1: How to Convert a Mixed Number to an Improper Fraction:

Multiple Choice

$$-\frac{10}{12}$$   $$-$$   $$\frac{3}{12}$$   $$=$$  

• $$\frac{7}{12}$$  

• $$-1\ \frac{1}{12}$$  

• $$\frac{6}{12}$$  

• $$-\frac{30}{144}$$  

Multiple Choice

$$\frac{10}{24}$$   $$+$$   $$\frac{6}{24}$$   $$=$$  Be sure to reduce your answer.

• $$\frac{10}{20}$$  

• $$\frac{4}{24}$$  

• $$\frac{16}{24}$$  

• $$\frac{2}{3}$$  

Adding & Subtracting Fractions & Mixed Numbers with UNLIKE Denominators ​

​

Steps:

1. ​Convert mixed numbers to improper fractions if needed

2. ​​Find a common denominator

3. ​Convert the numerators

4. ​Add or subtract the numerators

5. ​Leave the denominator the same

6. ​Simplify

​Examples:

Multiple Choice

$$\frac{1}{5}\ and\ \frac{5}{6}$$  what number could be a common denominator for these two fractions?

• 6

• 30

• 5

• 11

Multiple Choice

$$\frac{8}{9}-\frac{1}{3}$$  =

• 7/9

• 11/9

• -5/9

• 5/9

Multiple Choice

$$6\frac{7}{8}+\ 4\frac{3}{4}=$$  

• $$11\frac{5}{8}$$  

• $$11\frac{1}{4}$$  

• $$10\frac{10}{12}$$  

• $$10\frac{5}{6}$$  

​Multiplying Fractions & Mixed Numbers

​Steps:

1. ​Convert mixed numbers to fractions if you need to

2. ​Multiply straight across: numerator x numerator and denominator x denominator

3. ​Simplify

​Examples:

Multiple Choice

Find the product.

• 21/40

• -21/40

• 10/40

• -10/40

Multiple Choice

• 4 1/5
• -2 1/10
• -4 1/5
• 3 3/10

Dividing Fractions & Mixed Numbers​

​(keep, change, flip)

​Steps:

​We cannot divide fractions, so we have to change fraction division problems to multiplying by the reciprocal:

• ​Keep, Change, Flip

Multiple Choice

$$\frac{1}{6}\div2$$  

• $$\frac{1}{3}$$  

• $$\frac{2}{6}$$  

• $$\frac{1}{12}$$  

• $$12$$  

• $$3$$  

Multiple Choice

$$\frac{2}{5}\div6\ \frac{2}{5}$$  

• $$16$$  

• $$\frac{25}{64}$$  

• $$\frac{1}{16}$$  

• $$2\ \frac{14}{25}$$  

Simplifying Expressions​

​combine like terms

​Simplifying Expressions

• An algebraic ​expression contains variables, numbers, and at least one operation.

  • ​Example: n + 2

• ​Steps to simplifying expressions:

​1. Distribute if needed.

​2. Combine like terms. A like term must have the same variables raised to the same powers.

​

• ​Examples of like terms:

  • 5 and 2 = 7

  • ​6x and -4x = 2x

Multiple Choice

Simplify: 6b + 7b - 10

• 3b
• 13b - 10
• 13b² - 10
• 3b²

Multiple Choice

Simplify by combining like terms.
-14w + 8 + 9w - 2

• 23w + 6
• -23w + 6
• -6w + 7
• -5w + 6

​Distributive Property

​Multiply a value to all terms of an expression inside of parentheses. (the parentheses disappear)

Examples:

​

1.) a(b + c) = ab + ac

2.) 2(4 + 3) = 2(4) + 2(3) = 8 + 6 = 14 

3.) 5(x - 7) = 5x - 35

​

Multiple Choice

Which of the following is equivalent to 2(c - 8)?

• 2c - 16
• 2c + 16
• 2c - 8
• c - 16

​Subtracting Linear Expressions

• ​All you do is distribute the negative to each term inside of the parentheses.

• ​You can also think: Keep, Change, Change.

• ​Example:

Multiple Choice

(7a + 4) - (9a + 2)

• -2a + 6
• 2a + 6
• -2a + 2
• 2a + 2

Solving Equations​

get x by itself using inverse operations​

​Steps for Solving Equations

​1. Circle the variable and draw a line through the equal sign to sperate sides

​2. Distribute if needed

​3. Combine like terms ON THE SAME SIDE OF THE EQUAL SIGN if needed

​4. Undo the addition or subtraction

​5. Undo the multiplication or division

​6. Your answer should be x = (a number)

An equation is a statement that the values of two mathematical expressions are equal (indicated by the = sign)

Examples: 5x = 25

      2x + 4 = 8

​Solving a 2-Step Equation Example:

Multiple Choice

3y = -15

• -5
• 5
• -3
• 3

Multiple Choice

-4x + 1 = 21

• x=120
• x=5
• x=30
• x=-5

Multiple Choice

4a + 3 = 15

• 2
• 4
• 1
• 3

Inequalities​

Multiple Choice

Match the graph with its inequality.

• a > 11
• a< 11
•  a ≥ 11
•  a ≤ 11

​Solving Inequalities

​Solving Inequalities Using Adding and Subtracting: follow same steps as solving an equation.

​Solving Inequalities Using Multiplication and Division: follow same steps as solving an equation EXCEPT YOU HAVE TO REVERSE THE INEQUALITY SYMBOL WHEN YOU MULTIPY OR DIVIDE BY A NEGATIVE

Multiple Choice

4x - 4 ≥ -20

• x ≥ -6

• x ≥ -4

• x ≤ -4

• x ≤ -6

Multiple Choice

^x/₄ + 2 > -3

• x < -20

• x > -20

• x > 20

• x < 20

Multiple Choice

You flip an inequality symbol when you...

• subtract
• multiply and divide only
• multiple by a negative number
• multiple or divide by a negative number

Multiple Choice

16 - 4a > 8

• a<-2
• a>-2
• a<2
• a>2