Chapter 7 Summary Key Terms equation (7.1) Properties of Equality (7.2) solve an inequality (7.) Representing Daily Life Situations Using Picture Algebra Drawing a picture can be used to model a situation. In the picture, draw a rectangle or box to represent the unknown quantity. Combine the variable box with either boxes labeled with numbers or more boxes with the same variable to represent different amounts in relation to the variable. A picture can also be used to calculate a solution to the variable and each combination of boxes. Pictures can also be used to represent a mathematical sentence using operations and an equals sign. An equation is a mathematical sentence you create by placing an equals sign,, between two expressions. Eli, Dixon, and Derek sold fruit bars to raise money for a school trip. Eli sold three times as many fruit bars as Dixon, and Derek sold 14 more fruit bars than Dixon. The picture drawn, and the calculation shows how to determine how many fruit bars each boy sold. Let c represent fruit bars. Dixon : Eli : Derek : c c c c c 64 14 c 1 (3c) 1 (c 1 14) 64 c 1 14 64 c 0 c 10 Dixon sold 10 fruit bars, Eli sold 3 3 10 30 fruit bars, and Derek sold 10 1 14 24 fruit bars. Chapter 7 Summary 379
Understanding Equality and Using a Balance Model A representation can be shown on a balance. When subtraction of a quantity is done one one side, subtraction of the same quantity must occur on the other side to maintain a balance. The same process must also be done to both sides when other operations like addition, multiplication, and division are done on one side of the balance. Subtract two squares from each side, which leaves three rectangles on one side and 12 squares on the other side. Then divide each side by 3, which leaves one rectangle on one side and four squares on the other side: 1 rectangle 4 squares. 3x 1 2 14 x 4 380 Chapter 7 Solving Equations and Inequalities
Solving Two-Step Equations To solve a two-step equation, the variable must be isolated on one side of the equation. Perform the inverse operations in the reverse order of operations found in the original equation. To determine if your solution is correct, substitute the value of the variable back into the original equation. If the equation remains balanced, then the calculation performed is the solution to the equation. The solution for the equation b 1 19 2 for b is shown. Subtract first, and then multiply. Check the solution. b 1 19 2 b 1 19 2 19 2 2 19 30 1 19 2 b 6 6 1 19 2 ( b ) (6) 2 2 b 30 Solving Two-Step Equations Some equations require only two steps to isolate the variable and solve the equation. To isolate the variable, use the inverse operation of each operation contained within the equation. Verify the solution algebraically by substituting the value of the variable into the original equation to see if it makes the equation true. The solution of the two-step equation 4t 1 17 1 is shown. Subtract first, then divide. 4t 1 17 1 4t 1 17 2 17 1 2 17 4t 4 216 4 t 24 Verify the solution by substituting 24 for t in the original equation. 4(24) 1 17 0 1 1 1 Chapter 7 Summary 381
Using Two-Step Equations to Solve a Problem An equation can be written to represent a problem situation. When this occurs, variables must be defined, calculations are performed, and finally there is a verification of the solution. Kendra made three dozen breakfast rolls. She sold half of them at the bake sale and split the remaining rolls between her two brothers. The use of a two-step equation is shown to determine how many rolls each brother received. Kendra made three dozen rolls, or 36 rolls. She sold half, or 18 rolls. Let r represent the number of rolls each brother received. So, 36 18 1 2r. The solution for the equation is shown. Verify the solution: Each brother received nine breakfast rolls. 36 18 1 2r 36 2 18 18 1 2r 2 18 18 2 2r 2 9 r 36 0 18 1 2(9) 36 36 Solving Two-Step Equations Involving the Distributive Property Use the Distributive Property to simplify the equation before it can be solved. Then, apply the inverse operations to isolate the variable, and solve the equation. The steps for solving the equation 130 2 6(n 2 2) 220 are shown. 130 2 6(n 2 2) 220 130 2 6n 1 12 220 142 2 142 2 6n 220 2 142 26n 26 78 26 n 213 382 Chapter 7 Solving Equations and Inequalities
Using Two-Step Equations and the Distributive Property to Solve a Problem with Decimals or Percents Write an equation to represent a problem situation. Define your variables, solve, and verify your solution. Hong rented a bike to ride at the beach. The cost was $1 for the first hour and $ per hour after that. If he paid $27.0, you can determine how many hours Hong rented the bike using a two-step equation and the Distributive Property. Let h represent the number of hours for which the bike is rented. Then the equation is 27.0 1 1 (h 2 1). The solution to the equation is shown. Verify the solution. Hong rented the bike for 3. hours. 27.0 1 1 (h 2 1) 27.0 1 1 h 2 27.0 10 1 h 27.0 2 10 10 2 10 1 h 17.0 h 3. h 27.0 0 1 1 (3. 2 1) 27.0 0 1 1 17. 2 27.0 27.0 Graphing One-Variable Inequalities A number line can be used to represent inequalities. The graph of an inequality in one variable is the set of all points on a number line that make the inequality true. Any simple inequality can be represented on a number line by a ray whose starting point is an open or closed circle. A ray begins at a starting point and goes on forever in one direction. A closed circle means that the starting point is part of the solution. An open circle means that the starting point is not part of the solution. The graph of the inequality x. 8 is shown. 0 1 2 3 4 6 7 8 9 10 11 12 Chapter 7 Summary 383
Solving Simple Inequalities To solve an inequality means to determine the values of the variable that make the inequality true. The objective when solving an inequality is similar to the objective when solving an equation. The variable must be isolated on one side of the inequality symbol by using the operations of addition, subtraction, multiplication, and division. It is necessary to reverse the inequality symbol when you multiply or divide each side of an inequality by a negative value. The steps for solving the inequality 21 2 7x. 28 are shown. 21 2 7x. 28 21 2 21 2 7x. 28 2 21 27x. 7 27x 27, 7 27 x, 21 384 Chapter 7 Solving Equations and Inequalities