ALGEBRA

One-Step Linear Equations

A linear equation is an equation where two expressions are set equal to each other. The equation is in the form ax + b + c = 0, where a is a non-zero constant and b and c are constants. The exponent on a linear equation is always 1, and there is no more than one solution to a linear equation.

1. Addition Property of Equality: Add the same number to both sides of the equation.

Example with Numbers:

Example with Variables

x – 3 = 9

x = 12

x – a = b

x = a + b

2. Subtraction Property of Equality: Subtract the same number from both sides of the equation.

Example with Numbers:	Example with Variables
x + 3 = 9 x + 3 – 3 = 9 – 3 x = 6	x + a = b x + a – a = b – a x = b – a

3. Multiplication Property of Equality: Multiply both sides of the equation by the same number.

Example with Numbers:

Example with Variables

$\frac{x}{3}=9$

$\frac{x}{3}\times 3=9\times 3$

x = 27

$\frac{x}{a}=b$

$\frac{x}{a}\times a=b\times a$

x=ab

4. Division Property of Equality: Divide both sides of the equation by the same number.

Example with Numbers:

Example with Variables

3x = 9

$\frac{3x}{3}=\frac{9}{3}$

x=3

ax=b

$\frac{ax}{a}=\frac{b}{a}$

$x=\frac{b}{a}$

Two-Step Linear Equations

A two-step linear equation is in the form ax + b = c, where a is a non-zero constant and b and c are constants. There are two basic steps in solving this equation.

Step 1. Use addition and subtraction properties of an equation to move the variable to one side of the equation and all number terms to the other side of the equation.

Step 2. Use multiplication and division properties of an equation to remove the value in front of the variable.

Multi-Step Linear Equations

In these basic examples of linear equations, the solution may be evident, but these properties demonstrate how to use an opposite operation to solve for a variable. Using these properties, there are three steps in solving a complex linear equation.

Step 1. Simplify each side of the equation. This includes removing parentheses, removing fractions, and adding like terms.

Step 2. Use addition and subtraction properties of an equation to move the variable to one side of the equation and all number terms to the other side of the equation.

Step 3. Use multiplication and division properties of an equation to remove the value in front of the variable.

In Step 2, all of the variables may be placed on the left side or the right side of the equation. The examples in this lesson will place all of the variables on the left side of the equation.

When solving for a variable, apply the same steps as above. In this case, the equation is not being solved for a value, but for a specific variable.

Solving Linear Inequalities

A linear inequality is similar to a linear equation, but it contains an inequality sign (<, >, ≤, ≥). Many of the steps for solving linear inequalities are the same as for solving linear equations. The major difference is that the solution is an infinite number of values. There are four properties to help solve a linear inequality.

Addition Property of Inequality: Add the same number to both sides of the inequality.

Example:
x – 3 < 9
x – 3 + 3 < 9 + 3
x < 12

Subtraction Property of Inequality: Subtract the same number from both sides of the inequality.

Example:
x + 3 > 9
x + 3 – 3 > 9 – 3
x > 6

Multiplication Property of Inequality (when multiplying by a positive number): Multiply both sides of the inequality by the same number.

Example:

$\frac{x}{3}$

≥ 9

$\frac{x}{3} \hspace{0.5mm}\times3\hspace{0.5mm}\ge \hspace{0.5mm}9\hspace{0.5mm}\times\hspace{0.5mm}3$

x ≥ 27

Division Property of Inequality (when multiplying by a positive number): Divide both sides of the inequality by the same number.

Example:
3x ≤ 9

$\frac{3x}{3}\hspace{0.5mm}\leq \hspace{0.5mm}\frac{9}{3}$

x ≤ 3

Multiplication Property of Inequality (when multiplying by a negative number): Multiply both sides of the inequality by the same number.

Example:

$\frac{x}{-3}$

≥ 9

$\frac{x}{-3}\times\hspace{0.5mm}-3 \geq 9\hspace{0.5mm} \times -3\hspace{0.5mm}$

x ≤ -27

Estimating

Estimations are rough calculations of a solution to a problem. The most common use for estimation is completing calculations without a calculator or other tool. There are many estimation techniques, but this lesson focuses on integers, decimals, and fractions.

To round a whole number, round the value to the nearest ten or hundred. The number 142 rounds to 140 for the nearest ten and to 100 for the nearest hundred. The context of the problem determines the place value to which to round.
Keep In Mind in estimation is an educated guess at the solution to a problem

Other estimation strategies include the following:

Using friendly or compatible numbers
Using numbers that are easy to compute
Adjusting numbers after rounding

Real-World Integer Problems

The following five steps can make solving word problems easier:

1. Read the problem for understanding.

2. Visualize the problem by drawing a picture or diagram.

3. Make a plan by writing an expression to represent the problem.

4. Solve the problem by applying mathematical techniques.

5. Check the answer to make sure it answers the question asked.

Powers

We have explained in previous lessons that exponents imply an expression of repeated multiplication, otherwise known as a power. There are many rules associated with exponents. Recall that any number to a power of 0 will equal 1, or

. Additionally, the negative exponent rule reveals that numbers with a negative power are equal to a fraction.
The square of a number is the number raised to the power of 2. The square root of a number, when the number is squared, gives that number.

, so the square of 100 is 10, or

$\sqrt{100}=10$

. Perfect squares are numbers with whole number square roots, such as 1, 4, 9, 16, and 25.

The cube of a number is the number raised to the power of 3. The cube root of a number, when the number is cubed, gives that number.

, so the cube of 1,000 is 100, or

$\sqrt[3]{1000}=10$

. Perfect cubes are numbers with whole number cube roots, such as 1, 8, 27, 64, and 125.

	$\sqrt{1}=1$		$\sqrt{49}=7$
	$\sqrt{4}=2$		$\sqrt{64}=8$
	$\sqrt{9}=3$		$\sqrt{81}=9$
	$\sqrt{16}=4$		$\sqrt{100}=10$
	$\sqrt{25}=5$		$\sqrt{121}=11$
	$\sqrt{36}=6$		$\sqrt{144}=12$

	$\sqrt[3]{1}=1$		$\sqrt[3]{343}=7$
	$\sqrt[3]{8}=2$		$\sqrt[3]{512}=8$
	$\sqrt[3]{27}=3$		$\sqrt[3]{729}=9$
	$\sqrt[3]{64}=4$		$\sqrt[3]{1000}=10$
	$\sqrt[3]{125}=5$
	$\sqrt[3]{216}=6$

Scientific notation is a large or small number written in two parts. The first part is a number between 1 and 10. In these problems, the first digit will be a single digit. The number is followed by a multiple to a power of 10. A positive integer exponent means the number is greater than 1, while a negative integer exponent means the number is smaller than 1. Negative exponents are commonly used to represent small decimal numbers, but positive exponents can also be used to represent larger values.

For example, the population of the United States is about

$3 \times10^8$

, and the population of the world is about

$7 \times 10^9$

. The population of the United States is 300,000,000, and the population of the world is 7,000,000,000. The world population is about 20 times larger than the population of the United States.

Polynomials

A polynomial is an expression that contains exponents, as well as variables, constants, and operations. The exponents of the variables are only whole numbers, and there is no division by a variable. The operations are addition, subtraction, multiplication, and division. Constants are terms without a variable. A polynomial of one term is a monomial; a polynomial of two terms is a binomial; and a polynomial of three terms is a trinomial.

To add polynomials, combine like terms and write the solution from the term with the highest exponent to the term with the lowest exponent. To simplify, first rearrange and group like terms. Next, combine like terms.

To subtract polynomials, rewrite the second polynomial using an additive inverse. Change the minus sign to a plus sign, and change the sign of every term inside the parentheses. Then, add the polynomials.

Using Polynomial Identities

There are many polynomial identities that show relationships between expressions Addition and subtraction expressions can be squared, and the square of a binomial expression can be written in two different forms:

A simple way for solving a binomial is to remember to FOIL the expression. The square implies to repeat the multiplication of the operation. To solve the expression would require multiplying the First, Outside, Inside, and then Last variables which results in the relationship of the expressions.