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3.1 Solving Two-Step Equations Goal: Solve two-step equations. Example 1 Using Subtraction and Division to Solve Solve 4x 9 7. Check your solution. 4x 9 7 Notice in Example 1 that you isolate x by working backward. First you subtract from each side and then you divide. 4x 9 Write original equation. 7 Subtract 4x Simplify. 4x 16 Divide each side by x . 4x 9 7 ( 4 . Simplify. Answer: The solution is Check: from each side. ) 9 7 7 Write original equation. Substitute for x. . Checkpoint Solve the equation. Check your solution. 1. 3x 8 26 Copyright © Holt McDougal. All rights reserved. 2. 21 4x 7 Chapter 3 • Pre-Algebra Notetaking Guide 43 3.1 Solving Two-Step Equations Goal: Solve two-step equations. Example 1 Using Subtraction and Division to Solve Solve 4x 9 7. Check your solution. 4x 9 7 Notice in Example 1 that you isolate x by working backward. First you subtract from each side and then you divide. Write original equation. 4x 9 9 7 9 4x 16 4x 16 4 4 x 4 Subtract 9 from each side. Simplify. Divide each side by 4 . Simplify. Answer: The solution is 4 . 4x 9 7 Check: ( 4 4 ) 9 7 7 7 Write original equation. Substitute for x. Solution checks . Checkpoint Solve the equation. Check your solution. 1. 3x 8 26 6 Copyright © Holt McDougal. All rights reserved. 2. 21 4x 7 7 Chapter 3 • Pre-Algebra Notetaking Guide 43 Using Addition and Multiplication to Solve Example 2 x 3 Solve 4 1. Check your solution. x 4 1 3 x 4 3 Write original equation. 1 x 3 x 3 Add Simplify. ( ) Multiply each side by x . Simplify. Answer: The solution is Check: to each side. x 4 1 3 4 1 3 . Write original equation. Substitute for x. 1 . Checkpoint Solve the equation. Check your solution. x 4 3. 7 2 44 Chapter 3 • Pre-Algebra Notetaking Guide b 5 4. 8 3 Copyright © Holt McDougal. All rights reserved. Using Addition and Multiplication to Solve Example 2 x 3 Solve 4 1. Check your solution. x 4 1 3 Write original equation. x 4 4 1 4 3 x 3 3 x 3 3 3 3 ( ) x 9 Add 4 to each side. Simplify. Multiply each side by 3 . Simplify. Answer: The solution is 9 . Check: x 4 1 3 9 4 1 3 1 1 Write original equation. Substitute for x. Solution checks . Checkpoint Solve the equation. Check your solution. x 4 b 5 3. 7 2 4. 8 3 36 44 Chapter 3 • Pre-Algebra Notetaking Guide 55 Copyright © Holt McDougal. All rights reserved. Solving an Equation with Negative Coefficients Example 3 Solve 2 3x 17. Check your solution. 2 3x 17 2 3x Write original equation. 17 Subtract 3x Simplify. 3x 15 Divide each side by x . 2 3x 17 23 ( . Simplify. Answer: The solution is Check: from each side. ) 17 17 Write original equation. Substitute for x. . Checkpoint Solve the equation. Check your solution. 5. 3 2y 19 Copyright © Holt McDougal. All rights reserved. 6. 5 4 m Chapter 3 • Pre-Algebra Notetaking Guide 45 Example 3 Solving an Equation with Negative Coefficients Solve 2 3x 17. Check your solution. 2 3x 17 Write original equation. 2 3x 2 17 2 3x 15 Subtract 2 from each side. Simplify. 3x 15 3 3 x 5 Divide each side by 3 . Simplify. Answer: The solution is 5 . Check: 2 3x 17 ( 2 3 5 17 ) 17 17 Write original equation. Substitute for x. Solution checks . Checkpoint Solve the equation. Check your solution. 5. 3 2y 19 8 Copyright © Holt McDougal. All rights reserved. 6. 5 4 m 9 Chapter 3 • Pre-Algebra Notetaking Guide 45 3.2 Solving Equations Having Like Terms and Parentheses Goal: Solve equations using the distributive property. Writing and Solving an Equation Example 1 Baseball Game A group of five friends are going to a baseball game. Tickets for the game cost $12 each, or $60 for the group. The group also wants to eat at the game. Hot dogs cost $2.75 each and bottled water costs $1.25 each. The group has a total budget of $85. If the group buys the same number of hot dogs and bottles of water, how many can they afford to buy? Solution Let n represent the number of hot dogs and the number of bottles of water. Then 2.75n represents the cost of n hot dogs and 1.25n represents the cost of n bottles of water. Write a verbal model. Substitute. Combine like terms. Subtract each side. Simplify. Divide each side by . n from Simplify. Answer: The answer must be a whole number. Round down so the budget is not exceeded. The group can afford to buy hot dogs and bottles of water. 46 Chapter 3 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 3.2 Solving Equations Having Like Terms and Parentheses Goal: Solve equations using the distributive property. Example 1 Writing and Solving an Equation Baseball Game A group of five friends are going to a baseball game. Tickets for the game cost $12 each, or $60 for the group. The group also wants to eat at the game. Hot dogs cost $2.75 each and bottled water costs $1.25 each. The group has a total budget of $85. If the group buys the same number of hot dogs and bottles of water, how many can they afford to buy? Solution Let n represent the number of hot dogs and the number of bottles of water. Then 2.75n represents the cost of n hot dogs and 1.25n represents the cost of n bottles of water. Write a verbal model. Cost of Cost of Total Cost of tickets budget bottled water hot dogs 2.75n 1.25n 60 85 Substitute. 4n 60 85 Combine like terms. 4n 60 60 85 60 Subtract 60 from each side. Simplify. 4n 25 4n 4 25 4 n 1 6 4 Divide each side by 4 . Simplify. Answer: The answer must be a whole number. Round down so the budget is not exceeded. The group can afford to buy 6 hot dogs and 6 bottles of water. 46 Chapter 3 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Example 2 Solving Equations Using the Distributive Property Solve the equation. a. 24 6(2 x) Solution a. 24 b. b. 2(7 4x) 10 24 6(2 x) Write original equation. 24 Distributive property from each side. Simplify. Divide each side by x Simplify. 2(7 4x) 10 . Write original equation. 10 Distributive property 10 Add Simplify. Divide each side by x Copyright © Holt McDougal. All rights reserved. Subtract to each side. . Simplify. Chapter 3 • Pre-Algebra Notetaking Guide 47 Solving Equations Using the Distributive Property Example 2 Solve the equation. a. 24 6(2 x) Solution a. b. 2(7 4x) 10 24 6(2 x) Write original equation. 24 12 6x Distributive property 24 12 12 6x 12 36 6x 36 6 Simplify. 6x Divide each side by 6 . 6 6 x b. Simplify. 2(7 4x) 10 14 8x 10 14 8x 14 10 14 8x 24 8x Distributive property Add 14 to each side. Simplify. Divide each side by 8 . 8 x 3 Copyright © Holt McDougal. All rights reserved. Write original equation. 24 8 Subtract 12 from each side. Simplify. Chapter 3 • Pre-Algebra Notetaking Guide 47 Example 3 Combining Like Terms After Distributing Solve 6x 4(x 1) 14. 6x 4(x 1) 14 6x Write original equation. 14 Distributive property 14 Combine like terms. 14 Subtract Simplify. Divide each side by x from each side. . Simplify. Checkpoint Solve the equation. Check your solution. 48 1. 20 5(3 x) 2. 4y 14 3y 28 3. 3(6 2x) 12 4. 5x 2(x 3) 30 Chapter 3 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Example 3 Combining Like Terms After Distributing Solve 6x 4(x 1) 14. 6x 4(x 1) 14 Write original equation. 6x 4x 4 14 Distributive property 2x 4 14 Combine like terms. 2x 4 4 14 4 2x 10 2x Subtract 4 from each side. Simplify. 10 2 Divide each side by 2 . 2 x 5 Simplify. Checkpoint Solve the equation. Check your solution. 1. 20 5(3 x) 7 3. 3(6 2x) 12 5 48 Chapter 3 • Pre-Algebra Notetaking Guide 2. 4y 14 3y 28 6 4. 5x 2(x 3) 30 8 Copyright © Holt McDougal. All rights reserved. 3.3 Solving Equations with Variables on Both Sides Goal: Solve equations with variables on both sides. Solving an Equation with the Variable on Both Sides Example 1 Solve 5n 7 9n 21. 5n 7 9n 21 5n 7 Write original equation. 9n 21 7 7 Subtract 21 Simplify. 21 Subtract from each side. from each side. Simplify. Divide each side by n Answer: The solution is . Simplify. . An Equation with No Solution Example 2 Solve 3(2x 1) 6x. 3(2x 1) 6x Write original equation. 6x Distributive property Notice that this statement true because the number 6x . . The equation has As a check, you can continue solving the equation. The statement 6x Subtract Simplify. from each side. true, so the equation has . Copyright © Holt McDougal. All rights reserved. Chapter 3 • Pre-Algebra Notetaking Guide 49 3.3 Solving Equations with Variables on Both Sides Goal: Solve equations with variables on both sides. Example 1 Solving an Equation with the Variable on Both Sides Solve 5n 7 9n 21. 5n 7 9n 21 5n 7 5n 9n 21 5n 7 4n 21 7 21 4n 21 21 28 4n Write original equation. Subtract 5n from each side. Simplify. Subtract 21 from each side. Simplify. 28 4n 4 4 Divide each side by 4 . 7 n Simplify. Answer: The solution is 7 . Example 2 An Equation with No Solution Solve 3(2x 1) 6x. 3(2x 1) 6x Write original equation. 6x 3 6x Distributive property Notice that this statement is not true because the number 6x cannot be equal to 3 more than itself . The equation has no solution . As a check, you can continue solving the equation. 6x 3 6x 6x 6x 3 0 The statement 3 0 Subtract 6x from each side. Simplify. is not true, so the equation has no solution . Copyright © Holt McDougal. All rights reserved. Chapter 3 • Pre-Algebra Notetaking Guide 49 Example 3 Solving an Equation with All Numbers as Solutions Solve 4(x 2) 4x 8. 4(x 2) 4x 8 4x 8 Write original equation. Distributive property Notice that for all values of x, the statement 4x 8 is . The equation has . Checkpoint Solve the equation. Check your solution. 50 1. 3n 6 5n 20 2. 12 x 4(3x 1) 3. 3(2n 4) 2(3n 6) 4. 2x 7 2x 13 Chapter 3 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Example 3 Solving an Equation with All Numbers as Solutions Solve 4(x 2) 4x 8. 4(x 2) 4x 8 4x 8 4x 8 Write original equation. Distributive property Notice that for all values of x, the statement 4x 8 4x 8 is true . The equation has every number as a solution . Checkpoint Solve the equation. Check your solution. 1. 3n 6 5n 20 13 3. 3(2n 4) 2(3n 6) all numbers 50 Chapter 3 • Pre-Algebra Notetaking Guide 2. 12 x 4(3x 1) no solution 4. 2x 7 2x 13 5 Copyright © Holt McDougal. All rights reserved. Solving an Equation to Find a Perimeter Example 4 Geometry Find the perimeter of the square. x6 3x Solution 1. A square has four sides of equal length. Write an equation and solve for x. Write equation. Subtract from each side. Simplify. Divide each side by x Simplify. 2. Find the length of one side by substituting expression. ( ) 3x 3 for x in either Substitute for x and multiply. 3. To find the perimeter, multiply the length of one side by p . . Answer: The perimeter of the square is units. Checkpoint Find the perimeter of the square. 5. 3x 8 5x Copyright © Holt McDougal. All rights reserved. Chapter 3 • Pre-Algebra Notetaking Guide 51 Example 4 Solving an Equation to Find a Perimeter Geometry Find the perimeter of the square. x6 3x Solution 1. A square has four sides of equal length. Write an equation and solve for x. 3x x 6 Write equation. 3x x x 6 x 2x 6 2x Subtract x from each side. Simplify. 6 2 Divide each side by 2 . 2 x 3 Simplify. 2. Find the length of one side by substituting 3 for x in either expression. ( ) 3x 3 3 9 Substitute for x and multiply. 3. To find the perimeter, multiply the length of one side by 4 . 4 p 9 36 Answer: The perimeter of the square is 36 units. Checkpoint Find the perimeter of the square. 5. 3x 8 5x 80 units Copyright © Holt McDougal. All rights reserved. Chapter 3 • Pre-Algebra Notetaking Guide 51 Focus On Algebra Rewriting Equations and Formulas Use after Lesson 3.3 Goal: Rewrite literal equations and formulas. Vocabulary Literal equation Example 1 Solving a Literal Equation Solve ax bx c for x. Then use the solution to solve 5x 2x 12. Solution 1. Solve ax bx c for x. Write original equation. c When solving you calculate When a literala slope, be sure use equation, followtothe the x- steps and y-you would same coordinates of thea take when solving two points in the of specific equation same order. that form. x Distributive property Assume . Divide each side by . 2. Use the solution to solve 5x 2x 12. x Solution of literal equation Substitute Simplify. for a, Answer: The solution of 5x 2x 12 is for b, and for c. . Checkpoint Solve the literal equation for x. Then use the solution to solve the specific equation. x x 1. a b c; 1 3 4 52 Chapter 3 • Pre-Algebra Notetaking Guide 2. ax c bx; 5x 9 4x Copyright © Holt McDougal. All rights reserved. Focus On Algebra Rewriting Equations and Formulas Use after Lesson 3.3 Goal: Rewrite literal equations and formulas. Vocabulary An equation such as ax b c, in which coefficients and constants have been replaced by letters Literal equation Solving a Literal Equation Example 1 Solve ax bx c for x. Then use the solution to solve 5x 2x 12. Solution 1. Solve ax bx c for x. When solving you calculate When a literala slope, be sure use equation, followtothe the x- steps and y-you would same coordinates of thea take when solving two points in the of specific equation same order. that form. ax bx c Write original equation. (a b)x c Distributive property x c Assume (a b) ≠ 0 . Divide each side by (a b) . a b 2. Use the solution to solve 5x 2x 12. x c Solution of literal equation a b 12 Substitute 5 for a, 2 for b, and 12 for c. 5 2 4 Simplify. Answer: The solution of 5x 2x 12 is 4 . Checkpoint Solve the literal equation for x. Then use the solution to solve the specific equation. x x 1. a b c; 1 3 4 x a(b c); 16 52 Chapter 3 • Pre-Algebra Notetaking Guide 2. ax c bx; 5x 9 4x c x ; 1 ab Copyright © Holt McDougal. All rights reserved. Rewriting an Equation Example 2 Solve 3y 6x 12 for y. Solution 3y 6x 12 Write original equation. Add to each side. . Multiply each side by Use and simplify. Checkpoint Solve the equation for y. 3. 2y 8 14x Example 3 4. 18 6x 9y Rewriting and Using a Geometric Formula The area A of a rectangle is given by the formula A lw where l is the length and w is the width. a. Solve the formula for the width w. b. Use the rewritten formula to find the width of the rectangle shown. A 640 ft2 w I 40 ft Solution a. A lw Original formula. w Notice in Example 1 that you isolate x by A 640 ft2 w working backward. First you subtract I side 40 and ft from each then you divide. b. Substitute each side by for w and for . in the rewritten formula. Write rewritten formula. Substitute Simplify. for and for . Answer: The width is Copyright © Holt McDougal. All rights reserved. Chapter 3 • Pre-Algebra Notetaking Guide 53 Example 2 Rewriting an Equation Solve 3y 6x 12 for y. Solution 3y 6x 12 Write original equation. 3y 6x 12 Add 6x to each side. 1 1 (3y) (6x 12) 3 3 Multiply each side by 1 . y 2x 4 Use distributive property and simplify. 3 Checkpoint Solve the equation for y. 3. 2y 8 14x 4. 18 6x 9y 2 y 2 x 3 y 7x 4 Example 3 Rewriting and Using a Geometric Formula The area A of a rectangle is given by the formula A lw where l is the length and w is the width. a. Solve the formula for the width w. b. Use the rewritten formula to find the width of the rectangle shown. A 640 ft2 w I 40 ft Solution a. A lw w Notice in Example 1 that you isolate x by A 640 ft2 w working backward. First you subtract I side 40 and ft from each then you divide. A Original formula. Divide each side by I . I b. Substitute 640 for A and 40 for I in the rewritten formula. w A Write rewritten formula. I 64 0 Substitute 640 for A and 40 for I . 16 Simplify. 40 Answer: The width is 16 feet. Copyright © Holt McDougal. All rights reserved. Chapter 3 • Pre-Algebra Notetaking Guide 53 3.4 Solving Inequalities Using Addition or Subtraction Goal: Solve inequalities using addition or subtraction. Vocabulary Inequality: Solution of an inequality: Equivalent inequalities: Example 1 Writing and Graphing an Inequality Air Travel An airline allows passengers to carry on-board one piece of luggage. Luggage that exceeds 40 pounds cannot be carried on-board. Write an inequality that gives the weight of luggage that cannot be carried on-board. Solution Let w represent the weight of the luggage. Because the weight cannot exceed 40 pounds, the weight of luggage that cannot be carried on-board must be . Answer: The inequality is 0 54 . Draw the graph below. 10 20 30 40 50 60 70 80 Chapter 3 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 3.4 Solving Inequalities Using Addition or Subtraction Goal: Solve inequalities using addition or subtraction. Vocabulary An inequality is a statement formed by placing an Inequality: inequality symbol, such as < or >, between two expressions. The solution of an inequality with a variable is the Solution of set of all numbers that produce true statements an inequality: when substituted for the variable. Equivalent inequalities are inequalities that have Equivalent inequalities: the same solution. Example 1 Writing and Graphing an Inequality Air Travel An airline allows passengers to carry on-board one piece of luggage. Luggage that exceeds 40 pounds cannot be carried on-board. Write an inequality that gives the weight of luggage that cannot be carried on-board. Solution Let w represent the weight of the luggage. Because the weight cannot exceed 40 pounds, the weight of luggage that cannot be carried on-board must be greater than 40 pounds . Answer: The inequality is w > 40 . Draw the graph below. 0 54 10 20 30 40 50 60 70 80 Chapter 3 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Addition and Subtraction Properties of Inequality Words Adding or subtracting the same number on each side of an inequality produces an equivalent inequality. The addition and subtraction properties of inequality are also true for inequalities involving ≤ and ≥. Algebra If a < b, then a c < b c and a c < b c. If a > b, then a c > b c and a c > b c. Solving an Inequality Using Subtraction Example 2 Solve m 9 ≤ 12. Graph and check your solution. m 9 ≤ 12 m9 Write original inequality. ≤ 12 Subtract m≤ Simplify. Answer: The solution is m ≤ 1 0 1 2 from each side. 3 4 5 6 . Draw the graph below. 7 Check: Choose any number less than or equal to the number into the original inequality. m 9 ≤ 12 Write original inequality. ? 9 ≤ 12 Substitute 0 for m. 12 . Solving an Inequality Using Addition Example 3 You can read an inequality from left to right as well as from right to left. For instance, "2 is greater than x" can also be read "x is less than 2." Algebraically, this means that 2 x can also be written as x 2. . Substitute Solve 7 < x 11. Graph your solution. 7 < x 11 7 Write original inequality. < x 11 Add <x Simplify. < x, or Answer: The solution is Copyright © Holt McDougal. All rights reserved. 1 0 1 2 3 4 5 6 to each side. . Draw the graph below. 7 Chapter 3 • Pre-Algebra Notetaking Guide 55 Addition and Subtraction Properties of Inequality Words Adding or subtracting the same number on each side of an inequality produces an equivalent inequality. The addition and subtraction properties of inequality are also true for inequalities involving ≤ and ≥. Algebra If a < b, then a c < b c and a c < b c. If a > b, then a c > b c and a c > b c. Solving an Inequality Using Subtraction Example 2 Solve m 9 ≤ 12. Graph and check your solution. m 9 ≤ 12 Write original inequality. m 9 9 ≤ 12 9 Subtract 9 from each side. m≤ 3 Simplify. Answer: The solution is m ≤ 3 . Draw the graph below. 1 0 1 2 3 4 5 6 7 Check: Choose any number less than or equal to 3 . Substitute the number into the original inequality. m 9 ≤ 12 Write original inequality. ? 0 9 ≤ 12 ≤ 12 9 Solution checks . Solving an Inequality Using Addition Example 3 You can read an inequality from left to right as well as from right to left. For instance, "2 is greater than x" can also be read "x is less than 2." Algebraically, this means that 2 x can also be written as x 2. Substitute 0 for m. Solve 7 < x 11. Graph your solution. 7 < x 11 Write original inequality. 7 11 < x 11 11 Add 11 to each side. 4 <x Simplify. Answer: The solution is 4 < x, or x > 4 . Draw the graph below. Copyright © Holt McDougal. All rights reserved. 1 0 1 2 3 4 5 6 7 Chapter 3 • Pre-Algebra Notetaking Guide 55 Checkpoint Solve the inequality. Graph and check your solution. 1. m 7 < 13 1 0 1 2 3 4 5 6 7 1 0 1 2 3 4 5 6 7 1 0 1 2 3 4 5 6 7 1 0 1 2 3 4 5 6 7 2. a 4 ≥ 5 3. x 2 ≤ 3 4. 6 < z 7 56 Chapter 3 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Checkpoint Solve the inequality. Graph and check your solution. 1. m 7 < 13 m<6 1 0 1 2 3 4 5 6 7 5 6 7 5 6 7 6 7 2. a 4 ≥ 5 a≥1 1 0 1 2 3 4 3. x 2 ≤ 3 x≤5 1 0 1 2 3 4 4. 6 < z 7 1 < z or z > 1 1 56 Chapter 3 • Pre-Algebra Notetaking Guide 0 1 2 3 4 5 Copyright © Holt McDougal. All rights reserved. 3.5 Solving Inequalities Using Multiplication or Division Goal: Solve inequalities using multiplication or division. Multiplication Property of Inequality Words Multiplying each side of an inequality by a positive number produces an equivalent inequality. Multiplying each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality. The multiplication properties of inequality are also true for inequalities involving >, ≤, and ≥. Algebra If a < b and c > 0, then ac bc. If a < b and c < 0, then ac bc. Solving an Inequality Using Multiplication Example 1 m 4 Solve > 2. m > 2 4 Write original inequality. m 4 Multiply each side by . Reverse inequality symbol. p2 Simplify. m Checkpoint Solve the inequality. Graph your solution. t 5 b 8 1. < 3 9 Copyright © Holt McDougal. All rights reserved. 10 2. ≤ 1 11 12 13 14 15 16 9 8 7 6 5 4 3 2 Chapter 3 • Pre-Algebra Notetaking Guide 57 3.5 Solving Inequalities Using Multiplication or Division Goal: Solve inequalities using multiplication or division. Multiplication Property of Inequality Words Multiplying each side of an inequality by a positive number produces an equivalent inequality. Multiplying each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality. The multiplication properties of inequality are also true for inequalities involving >, ≤, and ≥. Algebra If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc. Solving an Inequality Using Multiplication Example 1 m 4 Solve > 2. m > 2 4 4 m 4 < m < Write original inequality. 4 p 2 Multiply each side by 4 . Reverse inequality symbol. 8 Simplify. Checkpoint Solve the inequality. Graph your solution. t 5 b 8 1. < 3 2. ≤ 1 t < 15 9 Copyright © Holt McDougal. All rights reserved. 10 11 12 13 b ≥ 8 14 15 16 9 8 7 6 5 4 3 2 Chapter 3 • Pre-Algebra Notetaking Guide 57 Division Property of Inequality Words Dividing each side of an inequality by a positive number produces an equivalent inequality. Dividing each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality. The division properties of inequality are also true for inequalities involving >, ≤, and ≥. a c a If a < b and c < 0, then c Algebra If a < b and c > 0, then b . c b . c Solving an Inequality Using Division Example 2 Solve 11t ≥ 33. 11t ≥ 33 11t Write original inequality. 33 Divide each side by . Reverse inequality symbol. Simplify. t Checkpoint Solve the inequality. Graph your solution. 3. 4y ≤ 36 3 58 4 5 Chapter 3 • Pre-Algebra Notetaking Guide 4. 3x > 12 6 7 8 9 10 9 8 7 6 5 4 3 2 Copyright © Holt McDougal. All rights reserved. Division Property of Inequality Words Dividing each side of an inequality by a positive number produces an equivalent inequality. Dividing each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality. The division properties of inequality are also true for inequalities involving >, ≤, and ≥. a b c c a b If a < b and c < 0, then > . c c Algebra If a < b and c > 0, then < . Solving an Inequality Using Division Example 2 Solve 11t ≥ 33. 11t ≥ 33 11t 11 Write original inequality. 33 ≤ 11 t ≤ 3 Divide each side by 11 . Reverse inequality symbol. Simplify. Checkpoint Solve the inequality. Graph your solution. 3. 4y ≤ 36 4. 3x > 12 y≤9 3 58 4 5 Chapter 3 • Pre-Algebra Notetaking Guide 6 7 x < 4 8 9 10 9 8 7 6 5 4 3 2 Copyright © Holt McDougal. All rights reserved. 3.6 Solving Multi-Step Inequalities Goal: Solve multi-step inequalities. Example 1 Writing and Solving a Multi-Step Inequality Charity Walk You are participating in a charity walk. You want to raise at least $500 for the charity. You already have $175 by asking people to pledge $25 each. How many more $25 pledges do you need? Solution Let p represent the number of additional pledges. Write a verbal model. p ≥ ≥ ≥ ≥ p≥ Answer: You need at least ≥ Substitute. Subtract each side. from Simplify. Divide each side by . Simplify. more $25 pledges. Checkpoint 1. Look back at Example 1. Suppose you wanted to raise at least $620 and you already have raised $380 by asking people to pledge $20 each. How many more $20 pledges do you need? Copyright © Holt McDougal. All rights reserved. Chapter 3 • Pre-Algebra Notetaking Guide 59 3.6 Solving Multi-Step Inequalities Goal: Solve multi-step inequalities. Example 1 Writing and Solving a Multi-Step Inequality Charity Walk You are participating in a charity walk. You want to raise at least $500 for the charity. You already have $175 by asking people to pledge $25 each. How many more $25 pledges do you need? Solution Let p represent the number of additional pledges. Write a verbal model. Additional Amount Minimum Money already p ≥ pledges per pledge desired amount raised 175 25p ≥ 500 Substitute. 175 25p 175 ≥ 500 175 25p ≥ 325 25p 325 ≥ 25 25 p ≥ 13 Subtract 175 from each side. Simplify. Divide each side by 25 . Simplify. Answer: You need at least 13 more $25 pledges. Checkpoint 1. Look back at Example 1. Suppose you wanted to raise at least $620 and you already have raised $380 by asking people to pledge $20 each. How many more $20 pledges do you need? at least 12 more $20 pledges Copyright © Holt McDougal. All rights reserved. Chapter 3 • Pre-Algebra Notetaking Guide 59 Solving a Multi-Step Inequality Example 2 x 9 < 7 3 x 9 3 Original inequality < 7 Add x < 3 to each side. Simplify. x3 p Multiply each side by . Reverse inequality symbol. Simplify. x Checkpoint Solve the inequality. Then graph the solution. x 4 2. 2x 9 < 25 3 4 5 6 3. 3 ≥ 2 7 8 9 10 4. 2 > 4 x 9 8 7 6 5 4 3 2 60 Chapter 3 • Pre-Algebra Notetaking Guide 3 4 5 6 7 8 9 10 6 7 8 9 10 x 2 5. 4 ≤ 9 3 4 5 Copyright © Holt McDougal. All rights reserved. Solving a Multi-Step Inequality Example 2 x 9 < 7 3 Original inequality x 9 9 < 7 9 3 x < 2 3 3 x3 > x > Add 9 to each side. Simplify. 3 p 2 Multiply each side by 3 . Reverse inequality symbol. 6 Simplify. Checkpoint Solve the inequality. Then graph the solution. x 4 2. 2x 9 < 25 3. 3 ≥ 2 x<8 3 4 5 6 7 x≥4 8 9 10 4. 2 > 4 x 3 4 5 60 Chapter 3 • Pre-Algebra Notetaking Guide 7 8 9 10 8 9 10 x 2 5. 4 ≤ 9 x > 6 9 8 7 6 5 4 3 2 6 x ≤ 10 3 4 5 6 7 Copyright © Holt McDougal. All rights reserved. 3 Words to Review Give an example of the vocabulary word. Inequality Solution of an inequality Equivalent inequalities Literal equation Review your notes and Chapter 3 by using the Chapter Review on pages 156–159 of your textbook. Copyright © Holt McDougal. All rights reserved. Chapter 3 • Pre-Algebra Notetaking Guide 61 3 Words to Review Give an example of the vocabulary word. Inequality x>0 Equivalent inequalities 2x < 10 and x > 5 Solution of an inequality The solution of m 9 ≤ 12 is m ≤ 3. Literal equation ax by c Review your notes and Chapter 3 by using the Chapter Review on pages 156–159 of your textbook. Copyright © Holt McDougal. All rights reserved. Chapter 3 • Pre-Algebra Notetaking Guide 61