Chapter Outline

Algebra

Algebra uses letters or symbols to represent unknown numbers or variables.

Working with Signed Numbers

The number line shows positive numbers to the right of zero and negative numbers to the left of zero.

• A number on the number line is greater than any number to its left.
• A number on the number line is less than any number to its right.
• Absolute value is the distance from a number to zero on the number line.

To add two signed numbers, follow these steps:

• If the signs are the same, add and give the total the sign of the numbers.
• If the signs are different, subtract and give the total the sign of the number with the greater absolute value.

To add more than two signed numbers, follow these steps:

• Add the positive numbers and make the sum positive.
• Add the negative numbers and make the sum negative.
• Find the difference between the two sums and give the answer the sign of the sum with the greater absolute value.

Subtracting a number means adding its opposite. On the number line opposite means "on the other side of zero." To subtract signed numbers, follow these steps:

• Change the sign of the number being subtracted and drop the subtraction sign.
• Follow the rules for adding signed numbers.

To multiply two signed numbers, follow these steps:

• Multiply the two numbers.
• If the signs of the two numbers are alike, make the product positive.
• If the signs of the two numbers are different, make the product negative.

To multiply more than two signed numbers, follow these steps:

• Multiply all the numbers together.
• If the problem has an even number of negative signs, the final product is positive.
• If the problem has an odd number of negative signs, the final product is negative.

To divide two signed numbers, follow these steps:

• Divide or reduce the numbers.
• If the signs are alike, make the quotient positive.
• If the signs are different, make the quotient negative.

Simplifying Algebraic Expressions

To simplify an expression, combine like terms:

• Combine x-terms with x-terms and numerical terms with numerical terms.
• Remember that x is the same as 1x.

To evaluate an expression, substitute a value for the unknown.

To simplify expressions with parentheses, use the distributive property:

• a(b + c) = ab + ac
• a(b - c) = ab - ac

Solving One-Step Equations

An equation is a statement that two amounts are equal. The = sign separates the two sides of an equation.

To solve an equation with one operation, perform the inverse, or opposite, operation on both sides of the equation:

• The inverse of addition is subtraction.
• The inverse of subtraction is addition.
• The inverse of multiplication is division.
• The inverse of division is multiplication.

Solving Longer Equations

To solve an equation with more than one operation, follow these steps:

• Add or subtract first.
• Then multiply or divide.

To solve an equation with separated unknowns, follow these steps:

• If the unknowns are on the same side of the = sign, follow the rules for adding and subtracting.
• If the unknowns are on different sides of the = sign, combine the unknowns using inverse operations.
• Use inverse operations to solve the equation.

To solve an equation with parentheses, follow these steps:

• Multiply each term inside the parentheses by the number outside the parentheses.
• Combine the unknowns and the numbers.

Solving Inequalities

An inequality is a statement that two amounts are not equal. There are four symbols used to write inequalities:

• < (is less than)
• > (is greater than)
• ≤ (is less than or equal to)
• ≥ (is greater than or equal to)

Writing Algebraic Expressions

To use algebra to solve word problems, you must translate mathematical relationships into algebraic expressions.

• Watch for words that suggest mathematical operations.
• Watch for verbs such as is and equals. These verbs tell you where to put the = sign.

Using Algebra to Solve Word Problems

Organize the information in word problems with algebraic expressions.

Using Algebra to Solve Geometry Problems

Substitute algebraic expressions into geometry formulas. Then solve for the unknowns.