Interactive Study Guide Solving Two-Step Equations

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1 11-1 To solve equations with more than one operation, or a two-step equation, follow the order of operations in reverse. First add or subtract then, multiply or divide. Solving Two-Step Equations Using Division Solve. Check your answer. 7x What number do you subtract from both sides? 7x 2 7 x What number do you divide by to isolate the variable? x What does x equal? 7x ( 4 ) 9? 37 Substitute 4 for x into the equation. 2 9? 37 Multiply Does the answer check? yes Solving Two-Step Equation Using Multiplication Solve. Check your answer. 9 p 1 9 Solving Two-Step Equations p What number do you subtract from both sides? ( ) p (7) What number do you multiply by to isolate the variable? 5 p 9 p 1 9 5? 1 Substitute 5 for p into the equation. 9 7? 1 Divide. 1 1 Does the answer check? yes Holt Middle School Math Course 2

2 11-2 Solving Multistep Equations There is usually more than one way to solve a multistep equation. There are steps you can take, however, to make the equation easier to solve. For instance, adding like terms on each side of the equation simplifies the solving process. Also, clearing denominators helps to avoid operating with fractions. Combining Like Terms Before Solving an Equation Solve 5x 12 1x 3. 5x 12 1x 3 4 x 12 3 Combine like terms on the left side of the equation Subtract 12 from both sides. 4 x 2 4 Divide both sides by 4 to isolate the variable. 4 4 x Solve for x. Clearing Denominators Before Solving an Equation Solve 3x x 3 4 ( ) 3x 3 4( ) Multiply both sides of the equation by the denominator. 3x 3 24 Multiply. 3x 3 24 What number do you subtract or add to both sides? 3 3x 21 3 x x 3 7 What number do you divide by to isolate the variable? What does x equal? 9 Holt Middle School Math Course 2

3 11-3 Solving Equations with Variables on Both Sides When solving equations with variables on both sides of the equation, get all of the variable terms on one side of the equation by adding or subtracting. Using Inverse Operations to Group Terms with Variables Group the terms with variables on one side of the equal sign and simplify. A. 4g 4 g 4g 4 g To get the variables on one side of the equation, 4g 4g what do you add or subtract from both sides? 4 2 g Combine like terms g 2 By what number do you divide both sides? 2 g Solve for g. B. 5d 11d 5d 11d To get the variables on one side of the equation, 5d 5d what do you add or subtract from both sides? 1 d Combine like terms. 1 1 d 1 By what number do you divide both sides? 1 2 d Solve for d. Consumer Application For Chrissy to do her laundry, she used to pay $10 per month to rent a washer and a dryer. However, Chrissy just purchased a new washing machine for $210. She still rents the dryer for $3 per month. How many months will it take for the amount it cost Chrissy to rent the washer and dryer to cost as much as it does for her to buy the new washing machine and rent only the dryer? 210 3t 10t Let t represent the number of months. 3t 3t What do you subtract from both sides? t Combine like terms t By what number do you divide both sides? t Solve for t. 2 1 After 30 months or 2 years the cost will be the same. 90 Holt Middle School Math Course 2

4 11-4 Inequalities An equation states that two values are equal. An inequality states that two values either are not equal or may not be equal. Where an equation uses the equal sign,, an inequality uses one of the following: less than less than or equal to greater than greater than or equal to Vocabulary inequality Writing Inequalities Write an inequality for each situation. A. There are at least runners on a track team. number of runners At least means greater than or equal to. B. There are fewer than 12 students in government. number of students 12 Fewer than means less than. C. No more than 50 people can swim in the pool. number of people 50 No more than means less than or equal to. Graphing Inequalities Graph each inequality. A. f Draw an circle on 3 because 3 included as a solution. Draw an arrow from 3 going to the smaller than 3 are all to the left of 3. B. k 1 open closed left of 3 because the numbers Draw a circle on 1 because 1 included as a solution. left Draw an arrow from 1 going to the smaller than 1 are all to the left of 1. is not is of 1 because the numbers 91 Holt Middle School Math Course 2

5 11-5 A solution set is the set of all the solutions of an inequality. Solutions of an inequality are the values that make the inequality true. Solving Inequalities by Adding Solve. Then graph the solution set on a number line. u 2 u 2 Solving Inequalities by Adding or Subtracting What number do you add to both sides to isolate the variable? u 4 Solve for u. Graph your answer right Draw an open circle on and an arrow going to the of. Vocabulary solution set solution of an inequality Solving Inequalities by Subtracting Solve. Check each answer. A. c 7 15 c What number do you subtract from both sides? c c 7 15 Substitute a number larger than in for c ? Does the solution check? yes B. b b What number do you subtract from both sides? b 5 b Substitute a number smaller than 5 in for b Does the solution check? yes 92 Holt Middle School Math Course 2

6 11- Solving Inequalities by Multiplying or Dividing When you multiply or divide both sides of an inequality by a negative number, the inequality symbol has to be reversed. Solving Inequalities by Multiplying Solve. d A ( 12 d ) 1 7( 12 ) Multiply both sides of the inequality by Since you multiplied by 12 do you need to reverse the inequality symbol? no d 4 Solve for d. h B ( 5 h ) 25( 5 ) Multiply both sides of the inequality by 5. 5 Since you multiplied by 5 do you need to reverse the inequality symbol? yes h 125 Solve for h. Solving Inequalities by Dividing Solve. Check your answer. x 3 x 3 Divide both sides of the inequality by. x Since you divided by do you need to reverse the yes inequality symbol? x 3 Substitute a number smaller than in for x. (0)? Does the solution check? yes 93 Holt Middle School Math Course 2