Solving One-Step Equations and Inequalities-Chapter 2

2-1 Properties of Numbers Objective 1: Identifying Properties

Commutative Properties of Additionand Multiplication

Changing the order of the values you are adding or multiplying does not change the sum or product.

Associative Properties of Addition and Multiplication

Changing the grouping of the values you are adding or multiplying does not change the sum or product.

Real-World    Problem Solving

Golf Carlos rented a set of golf clubs for $7 and a golf cart for $12. He paid a greens fee of $23. Find his total cost.

Use the associative property to solve this.

When you add a number and 0, the sum equals the original number. The additive identity is 0. When you multiply a number and 1, the product equals the original number. The multiplicative identity is 1.

Identity Properties of Addition and MultiplicationThe sum of any number and zero is the original number. The product of any number and 1 is the original number.

Identifying PropertiesName each property shown.

a. 5x7=7x5b. C x 1= cc. 7+a=a+7d. 5(xy)=(5x)y

Objective 2: Using PropertiesUsing Mental Math With Addition

Use mental math to simplify (81 + 6) + 9.

Real-World    Problem Solving

School Supplies Suppose you buy the school supplies shown at the left. Use mental math to find the cost of the supplies.

Using Mental Math With Multiplication

Use mental math to simplify (4 • 9) • 5.

2-2 The Distributive Property Objective 1: Numerical Expressions

Distributive PropertyTo multiply a sum or difference, multiply each number within the parentheses by the number outside the parentheses.

Use the Distributive Property to find 20(102) mentally. Using the Distributive Property I

Real-World     Problem Solving

Fundraising At the Parent-Teacher Association (PTA) Pancake Breakfast, the PTA served 397 people 4 pancakes each. How many pancakes did the PTA serve?

Using the Distributive Property II

Simplify 8(15) − 8(5).

Using Tiles to Multiply

Use algebra tiles to multiply 3(2x + 5).

Using the Distributive Property III

Multiply

a. -5(4x-3)

b. (2x+5)7

2-3 Simplifying Variable Expressions Objective 1: Identifying Partsof a variable expression

Identifying Parts of an Expression

Name the coefficients, the like terms, and the constants in 3m − 2n + n − 4.

Coefficients Like Terms Constants

Objective 2: Simplifying Variable ExpressionsYou simplify a variable expression by replacing it with an equivalent expression that has as few terms as possible. Algebra tiles can help you model this process.

Using Tiles to Simplify

Simplify 2x + 4 + 3x.

Combining Like Terms

Simplify 5y + y.

Deductive reasoning is the process of reasoning logically from given facts to a conclusion. As you use properties, rules, and definitions to justify the steps in a problem, you are using deductive reasoning.

2-4 Variables and Equations Objective 1: Classifying Types of Equations

An equation is a mathematical sentence with an equal sign. Here are three of the ways you will see equations in this book.

Classifying EquationsState whether each equation is true, false, or an open sentence.

a. 6+12=18b. 6=4+3c. 6y=-3+5y

An equation with a numerical expression equal to another numerical expression is either true or false. An equation with one or more variables is an open sentence.

Write an equation for Nine times the opposite of five is forty-five.

Writing an Equation

Objective 2: Checking Equations Using SubstitutionA solution of an equation is a value for a variable that makes an equation true. You substitute a number for a variable to determine whether the number is a solution of the equation.

Substituting to CheckIs 30 a solution of the equation 170 + x = 200?

Real-World    Problem Solving

Scuba Diving A diver's equipment weighs 35 lb. The diver plus the equipment weighs 165 lb. Can the diver's weight be 200 lb?

2-5 Solving Equations by Adding or Subtracting

Objective 1: Using Subtraction to Solve Equations

When you solve an equation, your goal is to get the variable alone on one side of the equation. The value on the other side tells you the solution of the original equation. You use inverse operations, which undo each other, to get the variable alone.

Subtraction Property of Equality

You can subtract the same number from each side of an equation.

Subtracting to Solve an EquationSolve x + 6 = 4

Real-World   Problem SolvingHealth Fred's target heart rate is 130 beats/min. This is 58 beats/min more than his resting heart rate. Find his resting heart rate.

Objective 2: Using Addition to Solve Equations

Addition Property of EqualityYou can add the same number to each side of an equation.

Adding to Solve an EquationSolve b − 12 = −49.

Real-World Problem Solving

Purchasing Your friend's VCR cost $328 less than her TV. Her VCR cost $179. How much did her TV cost?

2-6 Solving Equations by Multiplying or DividingObjective 1: Using Division to Solve Equations

Division Property of EqualityIf you divide each side of an equation by the same nonzero number, the two sides remain equal.

Real-World    Problem Solving

Statistics The United States population in 1998 was twice the population in 1943. Find the 1943 population in millions.

Let P =population in 1943

Dividing to Solve an EquationSolve 5r = −20.

Objective 2: Using Multiplication to Solve Equations

Multiplication Property of EqualityYou can multiply each side of an equation by the same number.

Multiplying to Solve an EquationSolve    = −3.

2-8 Inequalities and Their GraphsAn inequality is a mathematical sentence that contains >, <, ≥, ≤, or ≠. Some inequalities contain a variable. Any number that makes an inequality true is a solution of the inequality. For example, −4 is a solution of y ≥ −5 because −4 ≥ −5.

1. An Open dot means that the inequality has line2. A Closed dot means that the inequality has line

Objective 1: Graphing Inequalities

Graphing Solutions of InequalitiesGraph the solutions of each inequality on a number line.

y < 3

x > −1

a ≤ −2

−6 ≤ g

Objective 2: Writing InequalitiesWriting Inequalities to Describe Graphs

Write the inequality shown in each graph

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-1 0Real-World Problem Solving

Nutrition Food can be labeled low sodium only if it meets the requirement established by the federal government. Use the table to write an inequality for this requirement.

2-9 Solving One-Step Inequalities by Adding or SubtractingObjective 1: Solving Inequalities by Subtracting

Subtraction Property of InequalityYou can subtract the same number from each side of an inequality.

Subtracting to Solve an InequalitySolve each inequality. Graph the solutions. n + 8 ≥ 19

−26 > y + 14

Real-World Problem Solving

Computers Nearly 32 megabytes (MB) of memory are available for running your computer. If its basic systems require 12 MB, how much memory is available for other programs?

Objective 2: Using Addition to Solve Inequalities

You can add the same number to each side of an inequality.

Adding to Solve an InequalitySolve n − 15 < 3.

2-10 Solving One-Step Equations by Multiplying or Dividing

Engineering An elevator can carry up to 2,500 lb. Suppose the weight of an average adult is 150 lb. At most how many average-sized adults can safely ride the elevator at the same time?

Real-World Problem Solving

Multiplying to Solve an inequalitySolve  ≥ 7.