Presentation on theme: "Solving One-Step Equations and Inequalities-Chapter 2"— Presentation transcript:
1Solving One-Step Equations and Inequalities-Chapter 2
22-1 Properties of Numbers Objective 1: Identifying PropertiesCommutative Properties of Additionand MultiplicationChanging the order of the values you are adding or multiplying does not change the sum or product.
3Associative Properties of Addition and Multiplication Changing the grouping of the values you are adding or multiplying does not change the sum or product.
4Real-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.
5When 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 MultiplicationThe sum of any number and zero is the original number. The product of any number and 1 is the original number.
65x7=7x5 C x 1= c 7+a=a+7 5(xy)=(5x)y Identifying Properties Name each property shown.5x7=7x5C x 1= c7+a=a+75(xy)=(5x)y
7Objective 2: Using Properties Using Mental Math With AdditionUse mental math to simplify (81 + 6) + 9.
8Real-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.
9Using Mental Math With Multiplication Use mental math to simplify (4 • 9) • 5.
102-2 The Distributive Property Objective 1: Numerical ExpressionsDistributive PropertyTo multiply a sum or difference, multiply each number within the parentheses by the number outside the parentheses.Using the Distributive Property IUse the Distributive Property to find 20(102) mentally.
11Real-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?
12Using the Distributive Property II Simplify 8(15) − 8(5).
13Using Tiles to Multiply Use algebra tiles to multiply 3(2x + 5).
14Using the Distributive Property III Multiply-5(4x-3)(2x+5)7
15Coefficients Like Terms Constants 2-3 Simplifying Variable ExpressionsObjective 1: Identifying Partsof a variable expressionIdentifying Parts of an ExpressionName the coefficients, the like terms, and the constants in 3m − 2n + n − 4.Coefficients Like Terms Constants
16Objective 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 SimplifySimplify 2x x.
17Combining 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.
182-4 Variables and Equations Objective 1: Classifying Types of EquationsAn 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 EquationsState whether each equation is true, false, or an open sentence.6+12=186=4+36y=-3+5y
19Nine times the opposite of five is forty-five. Writing an EquationWrite an equation forNine times the opposite of five is forty-five.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 x = 200?
20Real-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?
212-5 Solving Equations by Adding or Subtracting Objective 1: Using Subtraction to Solve EquationsWhen 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 EqualityYou can subtract the same number from each side of an equation.
22Subtracting to Solve an Equation Solve x + 6 = 4Real-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.
23Objective 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.
24Real-World Problem Solving Purchasing Your friend's VCR cost $328 less than her TV. Her VCR cost $179. How much did her TV cost?
252-6 Solving Equations by Multiplying or Dividing Objective 1: Using Division to Solve EquationsDivision Property of EqualityIf you divide each side of an equation by the same nonzero number, the two sides remain equal.
26Real-World Problem Solving Statistics The United States population in 1998 was twice the population in Find the 1943 population in millions.Let P =population in1943
28Multiplication Property of Equality Objective 2: Using Multiplication to Solve EquationsMultiplication Property of EqualityYou can multiply each side of an equation by the same number.Multiplying to Solve an EquationSolve = −3.
292-8 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 InequalitiesAn Open dot means that the inequality has lineA Closed dot means that the inequality has lineGraphing Solutions of InequalitiesGraph the solutions of each inequality on a number line.y < 3x > −1a ≤ −2−6 ≤ g
30Objective 2: Writing Inequalities Writing Inequalities to Describe GraphsWrite the inequality shown in each graph-1 0Real-World Problem SolvingNutrition 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.
312-9 Solving One-Step Inequalities by Adding or Subtracting Objective 1: Solving Inequalities by SubtractingSubtraction Property of InequalityYou can subtract the same number from each side of an inequality.
32Subtracting to Solve an Inequality Solve each inequality. Graph the solutions.n + 8 ≥ 19−26 > y + 14
33Real-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?
34Adding to Solve an Inequality Objective 2: Using Addition to Solve InequalitiesYou can add the same number to each side of an inequality.Adding to Solve an InequalitySolve n − 15 < 3.
352-10 Solving One-Step Equations by Multiplying or Dividing
36Real-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 inequalitySolve ≥ 7.