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**Solving One-Step Equations and Inequalities-Chapter 2**

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**2-1 Properties of Numbers**

Objective 1: Identifying Properties Commutative Properties of Addition and Multiplication Changing the order of the values you are adding or multiplying does not change the sum or product.

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**Associative Properties of Addition and Multiplication**

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

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**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.

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**When you add a number and 0, the sum equals the original number**

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 Multiplication The sum of any number and zero is the original number. The product of any number and 1 is the original number.

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**5x7=7x5 C x 1= c 7+a=a+7 5(xy)=(5x)y Identifying Properties**

Name each property shown. 5x7=7x5 C x 1= c 7+a=a+7 5(xy)=(5x)y

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**Objective 2: Using Properties**

Using Mental Math With Addition Use mental math to simplify (81 + 6) + 9.

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**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.

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**Using Mental Math With Multiplication**

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

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**2-2 The Distributive Property**

Objective 1: Numerical Expressions Distributive Property To multiply a sum or difference, multiply each number within the parentheses by the number outside the parentheses. Using the Distributive Property I Use the Distributive Property to find 20(102) mentally.

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**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?

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**Using the Distributive Property II**

Simplify 8(15) − 8(5).

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**Using Tiles to Multiply**

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

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**Using the Distributive Property III**

Multiply -5(4x-3) (2x+5)7

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**Coefficients Like Terms Constants**

2-3 Simplifying Variable Expressions Objective 1: Identifying Parts of 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

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**Objective 2: Simplifying Variable Expressions**

You 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 x.

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**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.

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**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. 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. Classifying Equations State whether each equation is true, false, or an open sentence. 6+12=18 6=4+3 6y=-3+5y

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**Nine times the opposite of five is forty-five.**

Writing an Equation Write an equation for Nine times the opposite of five is forty-five. Objective 2: Checking Equations Using Substitution A 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 Check Is 30 a solution of the equation x = 200?

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**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?

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**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.

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**Subtracting to Solve an Equation**

Solve x + 6 = 4 Real-World Problem Solving Health 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.

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**Objective 2: Using Addition to Solve Equations**

Addition Property of Equality You can add the same number to each side of an equation. Adding to Solve an Equation Solve b − 12 = −49.

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**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?

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**2-6 Solving Equations by Multiplying or Dividing**

Objective 1: Using Division to Solve Equations Division Property of Equality If you divide each side of an equation by the same nonzero number, the two sides remain equal.

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**Real-World Problem Solving**

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

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**Dividing to Solve an Equation**

Solve 5r = −20.

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**Multiplication Property of Equality**

Objective 2: Using Multiplication to Solve Equations Multiplication Property of Equality You can multiply each side of an equation by the same number. Multiplying to Solve an Equation Solve = −3.

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**2-8 Inequalities and Their Graphs**

An 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. Objective 1: Graphing Inequalities An Open dot means that the inequality has line A Closed dot means that the inequality has line Graphing Solutions of Inequalities Graph the solutions of each inequality on a number line. y < 3 x > −1 a ≤ −2 −6 ≤ g

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**Objective 2: Writing Inequalities **

Writing Inequalities to Describe Graphs Write the inequality shown in each graph -1 0 Real-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.

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**2-9 Solving One-Step Inequalities by Adding or Subtracting**

Objective 1: Solving Inequalities by Subtracting Subtraction Property of Inequality You can subtract the same number from each side of an inequality.

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**Subtracting to Solve an Inequality**

Solve each inequality. Graph the solutions. n + 8 ≥ 19 −26 > y + 14

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**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?

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**Adding to Solve an Inequality**

Objective 2: Using Addition to Solve Inequalities You can add the same number to each side of an inequality. Adding to Solve an Inequality Solve n − 15 < 3.

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

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**Real-World Problem Solving**

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? Multiplying to Solve an inequality Solve ≥ 7.

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Use the Distributive Property to: 1) simplify expressions 2) Solve equations.

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