# Pre-Algebra – Study Guide

Integers

The definition of an integer is any whole number. This includes both negative and positive values in addition to zero. Some examples of an integer include: {-2, -1, 0, 1, 2}

Example:

Which of the following numbers are not integers?

Variables

A variable is a placeholder for an unknown number. This placeholder is generally a letter such as x or y, but could also be another symbol, such as ♥ or ◊.

Examples:

If I know 3 + ♠ = 5, then I can tell that ♠ = 2. I am using “♠” as a variable.

If I want to know “what number plus five equals thirteen?” Then I can write the equation . I am using x as my variable.

Properties of Algebra

Associate Property:

This property allows you to change the grouping of numbers in a sum or product.

Example:

Example:

Commutative Property:

This property allows you to switch the positions of numbers in a sum or product.

Example:

Example:

This property says that anything plus zero is equal to itself.

Example:

Identity Property of Multiplication:

This property says that anything times one is equal to itself

Example:

Distributive Property:

This property says that multiplying a sum within a set of parentheses by a number is the same as multiplying each of the terms inside the parentheses by that number.

Example:

Division Property of Inequalities:

Multiplying or dividing each side of an inequality by a positive number produces an equivalent inequality. Multiplying or dividing each side of an inequality by a negative number and reversing the direction of the inequality symbol also produces an equivalent inequality.

Examples:

Expressions and Equations

An algebraic expression is a mathematical “phrase” containing variables and numbers. Not that writing numbers directly next to variables or parentheses is shorthand for multiplication.
Examples: are algebraic expressions.

To evaluate an expression, you can “plug in” a value in place of the variable, and simplify.

Example: Evaluate for . [Plug in 4 where you see an “x”]

Parts of an expression that are added together are called terms. has one term. has two terms: , and .

The number at the beginning of each term is called a coefficient. In the expression , the coefficient of x is 3, and the coefficient of is -2.

An equation is a mathematical “sentence” formed by setting two expressions equal to one another. Equations have equal signs, while expressions do not.

Examples:  and are equations.

solution is a value of the variable that makes an equation true. In the equation the solution is . If we replace x with 3 in the equation, is a true statement.

Like Terms

“Like terms” are those which have the same variables in them.
For example, 3x and 4x are like terms. But 3x and 4y are not.

Practice: Are the following pairs like terms?

3 and 6x ⇒ No.
1 and 2 ⇒ Yes.
and ⇒ No.
and ⇒ Yes.
and ⇒ No.
7a^{2}4 ⇒ Yes.

You can combine like terms through addition or subtraction. For example: 3x means three x’s. 4x means four x‘s. Three x’s and four x‘s make seven x‘s, so .

Terms that are not like terms can’t be added or subtracted. For example, 2x means two x‘s. 6y means six y’s. Their sum, cannot be simplified.

Practice: Simplify the following expressions by combining the like terms.

When evaluating expressions, it is important to follow the order of operations (PEMDAS):

1. Parentheses
2. Exponents
3. Multiplication and Division

Example:

is correct
is incorrect

To remember the order of operations, you can use the acronym PEMDAS (Parentheses, Exponents, Multiplications, Division, Addition, Subtraction). It may be helpful to remember this with the phrase “Please Excuse My Dear Aunt Sally” or some other pneumonic device.

Example:

Do everything inside the parentheses before dealing with the rest of the expression.

⇒ Multiply first.
⇒ Then subtract.

Next do the exponent.

Then the multiplication.

Solving Equations

To solve an equation for a variable, you must get the variable by itself on one side of the equation while keeping the equation balanced.

Equations are balanced as long as their two sides are equal. If you do something to one side of the equation, you must do it to the other side as well.

One-Step Equations
One-step equations only involve one mathematical operations. Look at what is done to the variable, and do the opposite. Then rewrite the equation.

Example:
Solve the equation
⇒ To undo the +5, subtract 5 from both sides.

Two-Step Equations
Two-step equations involve two different mathematical operations, usually multiplication or division and addition or subtraction. Start by getting all of the constants on one side. Rewrite the equation after each step.

Example:
Solve the equation:
⇒ Add 1 to both sides to undo the -1.

⇒ Divide both sides by 3 to get x alone.

Multi-Step Equations
To simplify a multi-step equation, start by distributing across parentheses and combining like terms. If there are variables on both sides of the equation, add or subtract to get them on the same side.

Example:
Solve the equation:
⇒ Distribute the -1.
⇒ Combine like terms 10 + 1.
⇒ Add x to both sides

⇒ Add 5 to both sides.

Divide both sides by 4.

It’s easy to check your work when solving equations. Just plug your answer back into the original equation, and evaluate both sides to see if it makes a true statement.

Example:
Check that is a solution to
Plug in 4 for x ⇒
Simplify ⇒

Solving Single Equations with Multiple Variables

When an equation contains more than one variable, it cannot be solved for a single value. It can, however, be solved for one variable in terms of the other variables.

To solve an equation for a variable, you must get that variable by itself on one side of the equation (and everything else on the other side) while keeping the equation balanced.

Example: Distance is equal to rate times time. Solve this equation for T.

⇒ To get T alone, divide both side by R

We now have that time is equal to distance over rate.

Solving Inequalities

Like an equation, an inequality is a mathematical sentence. An inequality compares two expressions, instead of setting them equal to each other. One of the following symbols will always appear in place of the equals sign:

<       Less than
>       Greater than
≤       Less than or equal to
≥       Greater than or equal to

Example:  means that five more than two times x is greater than 13.

Inequalities can be solved the same way as equations except for one thing. If you divide or multiply by a negative number, you must flip the inequality symbol. As with equations, it is important to do the same thing to both sides.

Example:

⇒ Subtract 5.
⇒ Divide by 2.

This means that x can be any number greater than 4.

Example:

⇒ Add 2x to get all the x terms together.
⇒ Subtract 5.
⇒ Divide by -4. Flip the inequality symbol.

All numbers less than 2 satisfy the inequality.

Algebraic Order of Operations

If you get stuck solving an equation, it’s nice to have a set of “rules” to prompt you to the next step. We can use the order of operations (PEMDAS) in reverse, to figure out what to undo first. This is sometimes referred to as the algebraic order of operations. Remember that there are many correct ways to solve an equation. This is just one method you might find useful.

Example:

⇒ There are no subtraction or addition steps outside the parentheses, so we look at division and multiplication first. PEMDAS

⇒ Undo multiplication by dividing. PEMDAS

⇒ There are no more division or multiplication steps, so look for exponents. PEMDAS

⇒ Undo the exponent by taking a square root. PEMDAS

⇒ Now we undo the parentheses. Drop them and re-start at the beginning with subtraction. PEMDAS

⇒ Undo the subtraction by adding 5. PEMDAS

⇒ There are no more subtraction or addition steps. PEMDAS

⇒ Undo the multiplication by dividing. PEMDAS

Ratios and Percents

When two quantities are divided as a means of comparing them, the result is called a ratio. Ratios are written in a variety of different ways. Below are three different ways of writing the ratio “three to five.”

percentage is a fraction (or ratio) out of 100. In order to convert from a fraction to a percentage, first convert to a decimal by dividing the numerator by the denominator. Multiply that number by 100, and add a percent symbol.

Example: Write the ratio “three to five” as a percentage.

Example: If four of ten marbles in a jar are blue, find the percentage of blue marbles in the jar.

1. Write the ratio of blue marbles  to total marbles ⇒
2. Write the ratio as a decimal ⇒
3. Multiply the decimal by 100% ⇒

40% of the marbles are blue.

Rational Numbers

A rational number is any number that can be expressed as a ratio of two integers with a non-zero denominator.

Example: Which of the following numbers are not rational numbers?

are not rational numbers. The other numbers can be written as ratios of integers, as shown below:

Probability

Probability is the likelihood that an event will occur. It is often denoted as . For example, if there is a 30% chance it will rain today, I might write . In a situation where a set of outcomes are equally likely (such as drawing a card or flipping a coin), the probability of an outcome can be calculated as the ratio of possible to successes to the total number of possible outcomes.

Example: Suppose a marble is drawn at random from a jar containing 4 blue marbles and 6 red marbles. What is the probability of drawing a blue marble?

There are 10 marbles that could be drawn, and 4 of them are blue. So there are 4 possible “successes” and 10 total possible outcomes.

Example: Madam Tucket has a bucket with eight acorn-flavored biscuits and two biscuit-shaped squirrels that have decided to stow away. If Madam Tucket decides to pull a biscuit-shaped object out of her bucket, what is the probability that it will be a biscuit-shaped squirrel, assuming no biscuits are eaten by the squirrels?

Find the total number of objects that could be pulled from the bucket.
. Then calculate the ratio:

There is a 20% chance of pulling a biscuit-shaped squirrel out of the bucket.

Multiplication Rule

When calculating the probability of multiple independent events happening together, (such as drawing two cards from a deck, or flipping two coins), multiply the individual probabilities to find the total probability.

Example: A coin is flipped and a 6-sided die is rolled at the same time. What is the probability of getting heads and 3?

First we calculate the probability of each individual event.

There is 1 “heads” side, and 2 sides total on the coin.
There is 1 “3” side, and 6 sides total on the die.

The outcome of the coin flip does not affect the outcome of the die roll, so we can use the multiplication rule.

There is a chance that we will get both heads and a 3.

When calculating the probability of getting either of two mutually exclusive outcomes (outcomes that cannot happen at the same time), add the two individual probabilities to find the total probability.

Example: A bag of marbles contains 1 green, 3 red, and 2 blue marbles. One marble is drawn. What is the probability that is either green or blue?

First calculate the probability of each individual event.

There is 1 green marble in a bag of 6 marbles.
There are 2 blue marbles in the bag of 6 marbles.

The marble drawn cannot be both green and blue, so we can use the addition rule.

There is a chance that we will get either green or blue.

Mean, Median, and Mode

The mean of a set of data is the average value. The mean is found by adding up all of the values and dividing by the total number of entries.

The median of a set of data is the value that falls in the middle when the entries are arranged from lowest to highest value. If there are an even number of entries, the median is the average of the middle two values.

The mode of a set of data is the value that appears most frequently. It is possible to have more than one mode, if two values appear the same number of times.

Example: Find the mean, median, and mode of the following data

First arrange the numbers from least to greatest ⇒ {1, 3, 4, 6, 6}

The middle value is 4, so our median is 4.

The mode is 6, because it appears twice, which is more than any other value.

.

Prime Factorization

Prime numbers are positive integers that are not divisible by any integers besides one and themselves. Integers that are the product of prime factors are called Composite numbers. The number 1 is not considered to be prime or composite.

Example: Which of the following are prime numbers, and which are composite?
3, 21, 5, 16, 105, 17, 11, 4

3, 5, 17, and 11 are prime.
21, 16, 105, and 4 are composite.

If a number is composite, you can find its prime factorization. One way of doing this is to use a factor tree like the one shown below. Start with a number you want to factor and find one divisor to start. Write two branches showing how the number can be split into two factors.

We can tell 1092 is divisible by 2, because it ends in an even digit. Dividing 1092 by 2 gives us 546.

At each step, if the number at the end of a branch is prime, circle it. If the number is composite, then draw new branches showing how it can be divided into smaller factors.

When all the branches on your tree end in prime numbers, write the prime factorization, listing the factors in increasing order. . Check by multiplying all the prime factors together.

Example:  Write the prime factorizations of 21, 16 and 105.

It can be useful to recognize some of the more commonly used prime numbers at a glance. The first thirty prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97, 101, 103, 107, 109, 113

The Cartesian Coordinate System

On the Cartesian plane, there are two axes. The x-axis runs horizontally from left to right, and the y-axis runs vertically from bottom to top. The point at which the axes cross is called the origin. The coordinates of a point on the Cartesian plane are written (x, y), where is the origin. The x and y axes divide the plane into four quadrants. You can tell what quadrant a point (x, y) lies in by the signs of its coordinates.

Example:

The point (2, 2) lies in the first quadrant.
The point (-1, 4) lies in the second quadrant.
The point (-2, -3) lies in the third quadrant.
The point (3, -2) lies in the fourth quadrant.

Slope of a Line

The slope of a line is equal to the change in y over the change in x, or “rise over run” between two points. The slope of a line can be calculated using the formula:

and are two points on the line, and m is the slope.

Linear Equations

Linear equations are first-degree polynomial equations. The solutions (x, y) to a linear equation for ma line on the Cartesian plane. Linear equations can be written in any of the following forms:

Standard Form
Where a, b, and c are constants, and a and b are not both zero.

Example: Write in standard form.

Both the x and y terms need to be on the left side of the equation.
Subtract 3x from both sides.
This equation is in standard form.

Slope Intercept Form
Where m is the slope of the line, and b is the y-intercept.

Example: Put the following equation into slope intercept form.

To put a linear equation into slope-intercept form, solve it for y.
⇒ Subtract 3x from both sides.
⇒ Divide everything by 4.
⇒ Simplify
⇒ This equation is in slope-intercept form.

This form of a linear equation is especially useful in graphing the function. With the form m is the slope of the line and b is the y-intercept, or the y-value at which the line will cross the y-axis.

Point Slope Form

Given a point (x_{1},\ y_{1}) and a slope m, we can write the equation of the line through that point with the given slope. The simplest way to do this is to use point-slope form.

Example: Write the equation of the line with m = 2 that passes through (1, 6).

This is the equation in point slope form. It can now be rearranged to slope intercept form or standard form if needed.

Functions

A function is a mathematical rule that takes an input value and converts it to a single output. The input, also called the independent variable, is typically denoted x. The output, also called the dependent variable, is often denoted, y, or f(x).

Examples:

If a function f(x) is to be evaluated for a certain input, say x = 3, we can write f(3) to denote the value of the function at 3.

Example: If , then .

The Vertical Line Test

You can tell if an equation is a function by performing the vertical line test. Look at the graph of the equation. Imagine moving a vertical line back and forth across the graph. (You can use your pencil or the edge of a rule for this). If there is any point at which the vertical line passes through the graph of the equation more than once, then it is not a function.

Domain and Range

The domain of a function is the set of all possible inputs. (The x-values). For each x-value in the domain, there must be a valid output. Sometimes there are x-values that cannot be used, because plugging them into the function results in an undefined point. If all numbers can be used as an input, the domain is “all real numbers” or .

Example: The domain of is all .

If we tried to calculate , we would get , which is undefined.

The range of a function is the set of all possible outputs. (The y-values). For each value y in the range, there must exist a valid input x in the domain such that f(x) = y.

Example: The range of is all x ≥ 0.

Negative numbers are not included in the range because the square of any real number (positive or negative) is always positive.

Example: Find the domain and range of the function

The only input for x that vies an undefined y-value is zero, so the domain is all x ≠ 0. We can get very large outputs by plugging in small x, and small outputs by plugging in large x, but there is no x-value that when plugged into this equation that gives us zero. There are also no negative outputs, because is always positive, and 2 divided by a positive value is positive.

Pythagorean Theorem

A right triangle is a triangle with one 90° angle. The longest side is called the hypotenuse, and the two shorter sides are called legs of the triangle. For a right triangle with leg lengths a and b, and hypotenuse length c.

Distance Formula

The distance formula results directly from the Pythagorean Theorem. If we wish to calculate the distance between two points, and , we can plot these points and draw a triangle.

The distance between the two points is measured by the hypotenuse. The leg lengths are and .

Using the Pythagorean Theorem, we can write the equation:

Taking the square root of both sides gives use the distance formula.