Topic 1 Review

1-7 Absolute Value Equations

and Inequalities

9/8

What is absolute value?

Absolute value bars, |x|, return x as a positive integer no matter if x is negative or
positive.

Steps to Remember:

1.

Always isolate the absolute value expression, |x|

2a. If there are operations inside of the absolute value
bars, consider both cases (negative and positive)

OR

2b. If there are no operations on the inside of the
absolute value bars, solve for the variable

3. If you ever see an absolute value equal a negative
number: no solution

Step 1: Subtract 2 on both
sides to isolate the |x|

Step 2b: There are no
operations on the inside of
the absolute value bars, so
solve for the variable

X can be positive 5 and
negative 5

|-5| = 5 and |5| = 5

Step 1: The bars are already
isolated (no operations are
being done outside the |2x-3|)

Step 2a: There are operations
inside of the absolute value
bars, consider both cases
(negative and positive)

|-1| = 1 and |1| = 1

We must see both cases where
2x-3 = 1 and where 2x-3 = -1

Step 1: Subtract 8 on both sides
and then divide by 3 on both
side to isolate the |x+6|

Step 3: Once we isolated, we
see that the absolute value of
x+6 = -1. Absolute value will
never give you a negative
number: no solution

1-6 Compound Inequalities

9/6

Steps to Remember:

1. Distribute if you have ()
2. Combine Like Terms
3. Get all numbers and variables

on opposite sides

4. With inequalities, flip the

direction if you divide or multiply
by a negative number

Divide by
negative = flip
inequality sign

1-5 Solving Inequalities in One

Variable

9/1

Steps to Remember:

1. Distribute if you have ()
2. Combine Like Terms
3. Get all numbers and variables

on opposite sides

4. With inequalities, flip the

direction if you divide or multiply
by a negative number

Distribute by multiplying -4 by 3x and -1

Subtract both sides of the inequality by 4

Combine like terms by adding -12x + 6x

Divide both sides of the inequality by -6 to get x alone on the left side

Remember to flip the inequality sign to the other direction

Multiply both side of the inequality by -4 (flip the inequality sign)

On the left side the -4 cancels out. On the right side 6(-4) = -24

Add 3 to both sides of the inequality

Divide both sides of the inequality by 7 to get x by itself

Get all of your like terms with x to the left side and all numbers to the right side

Subtract 1.5 x from both sides and subtract 19 from both sides of the inequality

Study Tip
You also could have gotten all
the like terms with x to the
right side and the numbers to
the left side instead

1-4 Literal Equations and

Formulas

8/29

This section is all about rearranging
equations. We are still following all rules as
before (in lessons 1-2 and 1-3)

The only difference is that your solution will
be a rearranged equation with one or more
variables and numbers instead of one
integer.

Refer here if you forget about perimeter or area of common shapes.

●The perimeter of any shape is the sum of
all sides

●This rectangle does not have numerical
values on each side; instead it has
variables W for width and L for length

●We can still add variable. The perimeter
is found by adding L+L+W+W.

●P = L+L+W+W OR P=2L + 2W (combining
like terms)

●P is a variable chosen to stand for
perimeter, but any letter other than W or
L would work

Solve for L

P = 2W + 2L

P - 2W = 2L

P

Rearrange the formula for the perimeter of a rectangle. P= 2W + 2L

-2W -2W

2 2

2
-W = L

Solve for W

P = 2W + 2L

P - 2L = 2W

P

-2L -2L

2 2

2
-L = W

When given values to substitute in an equation, this can help solve for
other unknown values. Use what you know to figure out what you do not
know.

P = 22 in

5 in

Rearrange the formula to find the length: the width = 5 in & the perimeter = 22 in.

P= 2W + 2L
22 = 2(5) + 2L
22 = 10 + 2L

12 = 2L

6 = L

-10 -10

2 2

A half hour = 30 minutes. There are 60
minutes in an hour. 30/60 = 0.5. We use
this decimal for t.

She has 15 minutes. There are 60
minutes in an hour. 15/60 = 0.25. We
use this decimal for t.

2.5
0.25 = r

10 = r

1-3 Solving Linear Equations

8/23

Steps to Remember:

1. Distribute if you have ()
2. Combine Like Terms
3. Get all numbers and variables

on opposite sides

1.Distribute the -2 by multiplying it by x
and -4

2.Combine like terms by adding 3x and
4x

3.Get the numbers and variables to
opposite sides by adding 2x to both
sides and 10 to both sides

4.Combine like terms again on both
sides of the equal sign 7x+2x=9x and
8+9+10=27.

5.Divide both sides by 9 to get x alone

6.x = 3

1.Distribute the 1/2 by multiplying it by n
and -4

2.Combine like terms -2 - 7 = -9

3.Get the numbers and variables to
opposite sides by adding 2n to both
sides and 9 to both sides

4.Combine like terms again on both
sides of the equal sign 1/2n +2n = 5/2n
and 6+9 =15.

5.Divide both sides by 5/2 to get n alone

6.n = 6

It does not matter if you add 2n first
or 9 first as long as you do the same
thing to both sides

When you follow all steps and get two equal numbers on both sides, this
is an identity and is considered infinitely many solutions.

When you follow all steps and get two unequal numbers on both sides,
this is considered no solution.

1-2 Solving Linear Equations

8/18

Steps to Remember:

1. Distribute if you have ()
2. Combine Like Terms
3. Get all numbers and variables

on opposite sides

Distribution: when you have Parentheses ()

Pay attention to your signs:

a negative times a negative equals a positive
a negative times a positive equals a negative

STEP
1

Combine Like Terms

0.2x

3x

-x

15x

0.8x

_ 4_
3_x

14

25

-12

2
4

-6

Numbers multiplied by a
variable can be combined with
other numbers multiplied by a
variable

Numbers, fractions, and
decimals without a variable
can be combined together

Getting the sum or difference of the things that are alike

2x + 6x = 8x

4x - 3x = x

13 + 4 = 17

5 - 7 = -2

2_
5_

STEP
2

Get all letters (variables) on a different
side than the numbers (constants)

8x + 3 = 19

8x = 16

x = 2

STEP
3

To get the x by itself undo every operation that was done to it. X was multiplied by 8 and then added by 3

We will subtract by 3 on both sides of the equal sign
to get rid of the 3 on the left side

- 3

- 3

We will divide by 8 on both sides of the equal sign
to get rid of the 8 on the left side

8

8

Multiply 2 by everything in the parentheses
Add 24 to both sides
Subtract 8 from both sides
Divide both sides by 2

Here is a different method

Multiply by 3 on both sides
Divide by 2 on both sides
Subtract 4 from both sides

Subtract by 3 on both sides
Divide by 3 on both sides

We are using the
variable x to represent
the 1st of the three
consecutive numbers

X + 1 represents the
number right after x

X + 2 represents the
number right after x + 1
(two numbers after x)

Write everything on this example in your
notebook.

1-1 Operations on Real

Numbers

8/15

A set is a collection of objects such as numbers.

Set A looks like this A= {1,2,3,4,5,6,7,8,9,10}

P= {

}

An element of a set is an object that is in the set. Write a set by

listing the elements, enclosed in curly braces ("{" and "}").

A= {1,2,3,4,5,6,7,8,9,10}

A= {1,2,3,4,5,6,7,8,9,10}

A set B is a subset of set A if each element of B is also an

element of A.

Elements of P that are red hearts

B= {2,4,6,8,10} Elements of A that are even

numbers

C= {5,10} Elements of A that are multiples of 5

A= {1,2,3,4,5,6,7,8,9,10}

In the set A of numbers from 1 to 10, which elements are in both
the subset of even numbers, B, and the subset of multiples of 5, C?

B= {2,4,6,8,10}

C= {5,10}

A= {1,2,3,4,5,6,7,8,9,10}

In the set A of numbers from 1 to 10, which elements are in both
the subset of even numbers, B, and the subset of multiples of 5, C?

B= {2,4,6,8,10} even numbers in A

C= {5,10}multiples of 5 in A

The number 10 is the only number that is an element of both
subsets.

<- Least

Greatest ->

<- Middle

1.

What is an irrational number, and give one example of an irrational
number?

Any real number that cannot be expressed as the quotient of two integers p/q. For example, there is no number
among integers and fractions that equals Square root of √2.

2.

What is a rational number, and give one example of a rational number?

Any real number that can be expressed as the quotient of two integers p/q, where q is not equal to zero.

Natural numbers: 1,2,3,4,5,6…..

Whole numbers: 0,1,2,3,4,5,6….

Integers: -3,-2,-1,0,1,2,3….

Real numbers: All numbers that can be represented on a number line

Rational numbers: all decimals, fractions, (if you have a square root, it is
rational only if it is a perfect square)

What is this number?

Irrational, because square root
of 3= 1.73 (decimal); NOT a
perfect square

Rational, because square root of
9= 3 (integer); IS a perfect square
9

You cannot divide by zero