Numbers
Let's get reacquainted with the players in the game.
Whole Numbers
First, you have whole numbers. Whole numbers are also called "counting numbers" and include 0, 1, 2, 3, 4, 5, 6, . . . You see the pattern here.
Integers
Next on the roster are integers. Integers are all positive and negative whole numbers including zero: –3, –2, –1, 0, 1, 2, 3, . . .
Rational Numbers
Rational numbers, good old rational numbers. Rational numbers are all numbers that can be written as fractions 
, terminating decimals (0.75), and repeating decimals (0.666. . .).
Irrational Numbers
And rounding out the team are irrational numbers. Irrational numbers cannot be expressed as terminating or repeating decimals. Some examples: π or √2
Order of Operations
Most people remember the order of operations by using a mnemonic device such as PEMDAS, or Please Excuse My Dear Aunt Sally. These stand for the order in which operations are done:
Multiplication and division are done in the order that they appear from left to right. Addition and subtraction work the same way—left to right.
Parentheses also include any grouping symbol such as brackets [ ] and braces {}.
Examples
Example 1:
–5 + 2 × 8 =
–5 + 16 =
11
Example 2:
9 + (6 + 2 × 4) – 32 =
9 + (6 + 8) – 32 =
9 + 14 – 9 =
23 – 9 =
14
Absolute Value
The absolute value is the distance of a number from zero and is expressed by placing vertical bars on either side of the number. For example, |–5| is 5 because –5 is 5 spaces from zero. Most people simply remember that the absolute value of a number is its positive form.
Examples
|–39| = 39
|92| = 92
|–11| = 11
|987| = 987
Variables - Numbers and Letters Meet
Variables are letters that are used to represent numbers. Once you realize that these variables are just numbers in disguise, you'll understand that they must obey all the rules of mathematics, just like the numbers that aren't disguised. This can help you figure out what number the variable at hand stands for.'
Coefficients
When a number is placed next to a variable, indicating multiplication, the number is said to be the coefficient of the variable. For example,
8c 8 is the coefficient to the variable c.
6ab 6 is the coefficient to both variables a and b.
Like Terms
If two or more terms have exactly the same variable(s), they are said to be like terms.
The process of grouping like terms together performing mathematical operations is called combining like terms. It is important to combine like terms carefully, making sure that the variables are exactly the same.
Expressions and Equations
An expression is like a series of words without a verb. Take, for example, 3x + 5 or a – 3.
An equation is a statement that includes the "verb," in this case, an equal sign. To solve an algebraic equation with one variable, find the value of the unknown variable.
Rules for Working with Equations
- The equal sign separates an equation into two sides.
- Whenever an operation is performed on one side, the same operation must be performed on the other side.
- To solve an equation, first move all of the variables to one side and all of the numbers to the other. Then simplify until only one variable (with a coefficient of 1) remains on one side and one number remains on the other side.
Example
| 7x – 11 = 29 – 3x |
Move the variables to one side. |
| 7x – 11 + 3x = 29 – 3x + 3x |
Perform the same operation on both sides. |
| 10x – 11 = 29 |
Now move the numbers to the other side. |
| 10x – 11 + 11 = 29 + 11 |
Perform the same operation on both sides. |
| 10x = 40 |
Divide both sides by the coefficient. |
 |
Simplify. |
| x = 4 |
Cross Products
You can solve an equation that sets one fraction equal to another by finding cross products of the fractions. Finding cross products allows you to remove the denominators from each side of the equation. Multiply each side by a fraction equal to 1 that has the denominator from the opposite side. For example...
First multiply one side by 
and the other by 
. The fractions and both equal 1, so they don't change the value of either side of the equation.
The denominators are now the same. Now multiply both sides by the denominator and simplify.
ad = bc
This example demonstrates how finding cross products works. In the future, you can skip all the middle steps and just assume that is the same as ad = bc.
Examples
 |
Find cross products. |
| 36x = 6 × 12 |
| 36x = 72 |
| x = 2 |
 |
Find cross products. |
| 4x = 16x + 12 |
| –12x = 12 |
| x = –1 |
Checking Equations
After you solve an equation, you can check your answer by substituting your value for the variable into the original equation.
Example
We found that the solution for 7x – 11 = 29 – 3x is x = 4. To check that the solution is correct, substitute 4 for x in the equation:
7x – 11 = 29 – 3x
7(4) – 11 = 29 – 3(4)
28 – 11 = 29 – 12
17 = 17
This equation is true, so x = 4 is the correct solution!
Special Tips for Checking Equations on Exams
If time permits, check all equations. For questions that ask you to find the solution to an equation, you can simply substitute each answer choice into the equation and determine which value makes the equation correct.
Be careful to answer the question that is being asked. Sometimes, questions require that you solve for a variable and then perform an operation. For example, a question may ask the value of x – 2. You might find that x = 2 and look for an answer choice of 2. But because the question asks for the value of x – 2, the answer is not 2, but 2 – 2. Thus, the answer is 0.
Equations With More Than One Variable
Some equations have more than one variable. To find the solution of these equations, solve for one variable in terms of the other(s). Follow the same method as when solving single-variable equations, but isolate only one variable.
Example
| 3x + 6y = 24 |
To isolate the x variable, move 6y to the other side. |
| 3x + 6y – 6y = 24 – 6y |
| 3x = 24 – 6y |
 |
Then divide both sides by 3, the coefficient of x. |
| x = 8 – 2y |
Then simplify, solving for x in terms of y. |
Inequalities
An inequality is two numbers or expressions that are connected with an inequality symbol. Inequalities contain the greater than, less than, greater than or equal to, less than or equal, or not equal to symbols. When you solve the inequalities for x, you can figure out a range of numbers that your unknown is allowed to be.
Here are some examples of inequalities:
| 2 < 5 |
(2 is less than 5) |
| 9 > 3 |
(9 is greater than 3) |
| 4 ≤ 4 |
(4 is less than or equal to 4) |
| 2x + 5 ≠ 11 |
(2x added to 5 is not equal to 11) |
Now, there is one rule that you need to remember when dealing with inequalities: When you multiply or divide by a negative number, you need to reverse the sign.
Tip: Whenever you multiply or divide an inequality by a negative number, you need to reverse the inequality symbol.
Example
–5x + 3 > 28 can also be expressed as what inequality?
The goal here is to isolate your x. First, subtract 3 from both sides.
–5x + 3 – 3 > 28 – 3
–5x > 25
When you multiply or divide by a negative number, you need to reverse the sign. When you divide by –5, you get:
So, x < –5.
Keep Positive - Rearranging Inequalities to Produce Positive Coefficients
Here's a Trick to avoid having to worry about flipping the sign: Just move your terms in a manner such that you will end up with a positive coefficient on your variable.
Let's look at –5x + 3 > 28 again. Our x coefficient is negative, so you will add +5x to both sides.
–5x + 3 + 5x > 28 + 5x
3 > 28 + 5x
Next, subtract 28 from both sides:
3 – 28 > 28 + 5x – 28
3 – 28 > 5x
–25 > 5x
You divide both sides by 5 to yield:
–5 > x
Solving Inequalities
You can solve inequalities with variables just like you can solve equations with variables. Use what you already know about solving equations to solve inequalities. Like equations, you can add, subtract, multiply, or divide both sides of an inequality with the same number. In other words, what you do to one side of an inequality, you must do to the other side.
Example
2x + 3 < 1
| Subtract 3 from both sides of the inequality. |
2x + 3 – 3 < 1 – 3 |
| Simplify both sides of the inequality. |
2x < –2 |
| Divide both sides of the inequality by 2. |
 |
| Simplify both sides of the inequality. |
x < –1 |
The answer for this example is the inequality x < –1. There is an endless number of solutions because every number less than –1 is an answer. In this problem, the number –1 is not an answer because the inequality states that your answers must be numbers less than –1.
Tip: The answer to an inequality will always be an inequality. Because the answer is an inequality, you will have an infinite number of solutions.
Solving Inequalities Versus Solving Equations
Did you notice the similarity between solving equations and solving inequalities? Well, there are some major differences you need to be aware of.
Notice what happens when you multiply or divide an inequality by a negative number.
| 2 |
< |
5 |
| –2 · 2 |
< |
5 · –2 |
| –4 |
< |
–10 |
However, –4 is not less than –10. So, –4 < –10 is a false statement. To correct it, you would have to rewrite it as –4 > –10.
You can solve inequalities using the same methods that you use to solve equations with these exceptions:
- When you multiply or divide an inequality by a negative number, you must reverse the inequality symbol.
- The answer to an inequality will always be an inequality.
Using Variables to Express Relationships
The most important skill for solving word problems is being able to use variables to express relationships. This list will assist you in this by giving you some common examples of English phrases and their mathematical equivalents.
- "Increase" means add.
- "Decrease" means subtract.
Examples
Example 1:
A number increased by five = x + 5.
A number decreased by five = x – 5.
- "Less than" means subtract.
- "More than" means add.
Example 2:
Ten less than a number = x – 10.
Ten more than a number = x + 10.
- "Times" or "product" means multiply.
- "Divisible" or "quotient" means divide.
Example 3:
Three times a number = 3x.
Three is divisible by a number = 3 ÷ x.
- "Times the sum" means to multiply a number by a quantity.
Example 4:
Five times the sum of a number and three = 5(x + 3).
- Two variables are sometimes used together.
Example 5:
A number y exceeds five times a number x by ten.
y = 5x + 10
- Inequality signs are used for "at least" and "at most," as well as "less than" and "more than."
Example 6:
The product of x and 6 is greater than 2.
x · 6 > 2
When 14 is added to a number x, the sum is less than 21.
x + 14 < 21
The sum of a number x and four is at least nine.
x + 4 ≥ 9
When seven is subtracted from a number x, the difference is at most four.
Absolute Value Inequalities
|x| < a is equivalent to –a < x < a, and |x| > a is equivalent to x > a or x < –a.
Example
|x + 3| > 7
x + 3 >7 or x + 3 < –7
x > 4 x < –10
Thus, x > 4 or x < –10.
Find practice problems and solutions for these concepts at Algebra Review Practice Problems.
Celebrate Memorial Day! Worksheets and Activities About American History 
May Workbooks are Here!
Get Outside! 10 Playful Activities
Add your own comment