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Mathematics: Applications and Concepts, Course 3 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 Lesson 10-1 Simplifying Algebraic Expressions Lesson 10-2 Solving Two-Step Equations Lesson 10-3 Writing Two-Step Equations Lesson 10-4 Solving Equations with Variables on Each Side Lesson 10-5 Inequalities Lesson 10-6 Solving Inequalities by Adding or Subtracting Lesson 10-7 Solving Inequalities by Multiplying or Dividing Example 1 Write Equivalent Expressions Example 2 Write Equivalent Expressions Example 3 Write Expressions with Subtraction Example 4 Write Expressions with Subtraction Example 5 Identify Parts of an Expression Example 6 Simplify Algebraic Expressions Example 7 Simplify Algebraic Expressions Example 8 Simplify Algebraic Expressions Example 9 Translate Phrases into Expressions Use the Distributive Property to rewrite Simplify. Answer: . Use the Distributive Property to rewrite Answer: Use the Distributive Property to rewrite Simplify. Answer: Use the Distributive Property to rewrite Answer: Use the Distributive Property to rewrite Rewrite Distributive Property Simplify. Definition of subtraction Answer: Use the Distributive Property to rewrite Answer: Use the Distributive Property to rewrite Rewrite Distributive Property Simplify. Answer: Use the Distributive Property to rewrite Answer: Identify the terms, like terms, coefficients, and constants in Definition of subtraction Identity Property; Answer: The like terms are The coefficients are The constant is –5. Identify the terms, like terms, coefficients, and constants in Answer: The terms are The like terms are The coefficients are The constant is –2. Simplify 6n – n. 6n and n are like terms. Identity Property; Distributive Property Simplify. Answer: Simplify 7n n. Answer: Simplify 5s 3 – 12s. are like terms. Commutative Property Distributive Property Answer: Simplify 6s 2 – 10s. Answer: Simplify 8z z – 5 – 9z 2. are like terms. –5 and 2 are also like terms. Definition of subtraction Commutative Property Distributive Property Simplify. Answer: –3 Simplify 6z z – 2 – 8z 2. Answer: –z THEATER Tickets for the school play cost $5 for adults and $3 for children. A family has the same number of adults as children. Write an expression in simplest form that represents the total amount of money spent on tickets. If x represents the number of adult tickets, then x also represents the number of children tickets. To find the total amount spent, multiply the cost of each ticket by the number of tickets purchased. Then add the expressions. Distributive Property Simplify. Answer: The expression $8x represents the total amount of money spent on tickets, where x is the number of adults or children. MUSEUM Tickets for the museum cost $10 for adults and $7.50 for children. A group of people have the same number of adults as children. Write an expression in simplest form that represents the total amount of money spent on tickets to the museum. Answer: $17.50x Example 1 Solve a Two-Step Equation Example 2 Solve Two-Step Equations Example 3 Solve Two-Step Equations Example 4 Equations with Negative Coefficients Solve Method 1 Use a model. Remove 1-tile from the mat. Separate the remaining tiles into 5 equal groups. There are 5 tiles in each group. Method 2 Use symbols. Use the Subtraction Property of Equality. Write the equation. Subtract 1 from each side. Use the Division Property of Equality. Divide each side by 5. Simplify. Answer: The solution is 5. Solve Answer: 6 Solve Check your solution. Method 1 Vertical Method Write the equation. Add 8 to each side. Simplify. Divide each side by 2. Simplify. Method 2 Horizontal Method Write the equation. Add 8 to each side. Simplify. Divide each side by 2. Simplify. Check Write the equation. Replace n with 21. The sentence is true. Answer: The solution is 21. Solve Answer: 9 Check your solution. Solve Write the equation. Subtract 2 from each side. Simplify. Multiply each side by 3. Simplify. Answer: The solution is –18. Solve Answer: –26 Solve Write the equation. Definition of subtraction Subtract 8 from each side. Simplify. Divide each side by –3. Simplify. Answer: The solution is –2. Solve Answer: –3 Solve Check your solution. Write the equation. Identity Property; Combine like terms; Add 2 to each side. Simplify. Divide each side by 2. Simplify. Check Write the equation. Replace k with 8. Multiply. The statement is true. Answer: The solution is 8. Solve Answer: 5 Example 1 Translate Sentences into Equations Example 2 Translate Sentences into Equations Example 3 Translate Sentences into Equations Example 4 Translate and Solve an Equation Example 5 Write and Solve a Two-Stop Equation Translate three more than half a number is 15 into an equation. Answer: Translate five more than one-third a number is 7 into an equation. Answer: Translate nineteen is two more than five times a number into an equation. Answer: Translate fifteen is three more than six times a number into an equation. Answer: Translate eight less than twice a number is –35 into an equation. Answer: Translate six less than three times a number is –22 into an equation. Answer: Two more than Words Variable Equation of a number is 6. Find the number. Two more than of a number is 6. Write the equation. Subtract 2 from each side. Simplify. Mentally multiply each side by 3. Answer: The number is 12. Three more than six times a number is 15. Find the number. Answer: 2 TRANSPORTATION A taxi ride costs $3.50 plus $2 for each mile traveled. If Jan pays $11.50 for the ride, how many miles did she travel? Her cost starts at $3.50 and adds $2 until it reaches $11.50. Organize the data for the first few miles into a table and look for a pattern. Miles 0 1 2 3 Cost Write an equation to represent the situation. Let m represent the number of miles. flat rate plus m miles at $2 per mile 3.50 + 2m equals $11.50 11.50 Write the equation. Subtract 3.50 from each side. Simplify. Divide each side by 2. Simplify. Answer: Jan traveled 4 miles. TRANSPORTATION A rental car costs $100 plus $0.25 for each mile traveled. If Kaya pays $162.50 for the car, how many miles did she travel? Answer: 250 miles Example 1 Equations with Variables on Each Side Example 2 Equations with Variables on Each Side Example 3 Use an Equation to Solve a Problem Solve Check your solution. Write the equation. Subtract 7x from each side. Simplify by combining like terms. Mentally divide each side by 2. To check your solution, replace x with 2 in the original equation. Check Write the equation. Replace x with 2. The sentence is true. Answer: The solution is 2. Solve Answer: ─3 Check your solution. Solve Write the equation. Subtract 8x from each side. Simplify. Add 2 to each side. Simplify. Mentally divide each side by –5. Answer: The solution is –3. Solve Answer: –10 GRID-IN TEST ITEM Find the value of x so that the polygons have the same perimeter. Read the Test Item You need to find the value of x that will make the perimeter of the triangle equal to the perimeter of the rectangle. Solve the Test Item Write expressions for the perimeter of each figure. Then set the two expressions equal to each other and solve for x. Triangle Rectangle Answer: GRID-IN TEST ITEM Find the value of x so that the polygons have the same perimeter. Answer: Example 1 Write Inequalities with < or > Example 2 Write Inequalities with < or > Example 3 Write Inequalities with or Example 4 Write Inequalities with or Example 5 Determine the Truth of an Inequality Example 6 Determine the Truth of an Inequality Example 7 Graph an Inequality Example 8 Graph an Inequality SPORTS Members of the little league team must be under 14 years old. Write an inequality for the sentence. Answer: SPORTS Members of the peewee football team must be under 10 years old. Write an inequality for the sentence. Answer: CONSTRUCTION The ladder must be over 30 feet tall to reach the top of the building. Write an inequality for the sentence. Answer: CONSTRUCTION The new building must be over 300 feet tall. Write an inequality for the sentence. Answer: POLITICS The president of the United States must be at least 35. Write an inequality for the sentence. Answer: VOTING To vote, you must be at least 18 years old. Write an inequality for the sentence. Answer: CAPACITY A theater can hold a maximum of 300 people. Write an inequality for the sentence. Answer: CAPACITY A football stadium can hold a maximum of 10,000 people. Write an inequality for the sentence. Answer: For the given value, state whether the inequality is true or false. Write the inequality. Replace x with 0. Simplify. Answer: Since –4 is less than 6, For the given value, state whether the inequality is true or false. Answer: false For the given value, state whether the inequality is true or false. Write the inequality. Replace x with 1. Simplify. Answer: Since 3 is not greater than or equal to 4, the sentence is false. For the given value, state whether the inequality is true or false. Answer: true Graph n –1 on a number line. Place a closed circle at –1. Then draw a line and an arrow to the left. Answer: The closed circle means the number –1 is included in the graph. Graph n Answer: –3 on a number line. Graph n –1 on a number line. Place an open circle at –1. Then draw a line and an arrow to the right. Answer: The open circle means –1 is not included in the graph. Graph n Answer: –3 on a number line. Example 1 Solve an Inequality Using Addition Example 2 Solve an Inequality Using Subtraction Example 3 Graph the Solutions of an Inequality Example 4 Use an Inequality to Solve a Problem Solve Check your solution. Write the inequality. Add 4 to each side. Simplify. Check Write the inequality. Replace n with a number greater than 10, such as 11. The statement is true. Answer: Any number greater than 10 will make the statement true, so the solution is Solve Answer: Check your solution. Solve Check your solution. Write the inequality. Subtract 8 from each side. Simplify. Check Replace x in the original inequality with –15 and then with a number less than –15. Answer: The solution is Solve Answer: Check your solution. Solve Then graph the solution on a number line. The solution is Graph the solution. Place an open circle at the left. Answer: Draw a line and an arrow to Solve number line. Answer: Then graph the solution on a TOWING COMPANY A pickup truck is towing a trailer that weighs 3,525 pounds. The maximum towing capacity of the truck is 4,700 pounds. Determine how much more weight can be added to the trailer and still be towed by the truck. Words The phrase maximum capacity means less than or equal to. So, the current weight being towed plus any more weight must be less than or equal to 4,700 pounds. Variable Inequality current weight plus 3,525 + must be weight less than or 4,700 added equal to pounds w 4,700 Write the inequality. Subtract 3,525 from each side. Simplify. Answer: Up to 1,175 more pounds can be added to the trailer. SPORTS A weightlifter can lift up to 375 pounds. He is currently lifting 255 pounds. Determine how much more weight can be added and still be lifted by the weightlifter. Answer: Up to 120 more pounds can be added. Example 1 Divide by a Positive Number Example 2 Multiply by a Positive Number Example 3 Multiply or Divide by a Negative Number Example 4 Multiply or Divide by a Negative Number Example 5 Solve a Two-Step Inequality Solve Check your solution. Write the inequality. Divide each side by 6. Simplify. Answer: The solution is You can check this solution by substituting numbers less than –5 into the inequality. Solve Answer: Check your solution. Solve and check your solution. Then graph the solution on a number line. Write the inequality. Multiply each side by 2. Simplify. The solution is You can check this solution by substituting 18 and a number greater than 18 into the inequality. Graph the solution, Answer: Solve and check your solution. Then graph the solution on a number line. Answer: Solve Check your solution. Write the inequality. Multiply each side by –4 and reverse the inequality symbol. Simplify. Answer: The solution is You can check this solution by replacing b in the original inequality with –20 and a number greater than –20. Solve Answer: Check your solution. Solve number line. Then graph the solution on a Write the inequality. Divide each side by –4 and reverse the inequality symbol. Check this result. Graph the solution, Answer: Solve number line. Answer: Then graph the solution on a PACKAGES A box weighs 1 pound. It is filled with books that weigh 2 pounds each. Jesse can carry at most 20 pounds. Assuming space is not an issue, write and solve an inequality to find how many books he can put in the box and still carry it. The phrase at most means less than or equal to. of books he puts in the box. Then write an inequality. 1 pound 1 plus 2 pounds per book 2p is less than or equal to 20 pounds 20 Write the inequality. Subtract 1 from each side. Simplify. Divide each side by 2. Simplify. Answer: Since he can not put half a book in the box, Jesse can put at most 9 books in the box. PACKAGES A box weighs 2 pounds. It is filled with toys that weigh 1 pound each. Danielle can carry at most 30 pounds. Assuming space is not an issue, write and solve an inequality to find how many toys she can put in the box and still carry it. Answer: She can put at most 28 toys in the box. Explore online information about the information introduced in this chapter. Click on the Connect button to launch your browser and go to the Mathematics: Applications and Concepts, Course 3 Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to www.msmath3.net/extra_examples. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. End of Custom Shows WARNING! Do Not Remove This slide is intentionally blank and is set to auto-advance to end custom shows and return to the main presentation.