1.9 Solving First-Degree Inequalities

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1 1.9 Solving First-Degree Inequalities Canadian long-track speed skater Catriona LeMay Doan broke world records in both the 500-m and the 1000-m events on the same day in Calgary. Event 500-m 1000-m Catriona s Time (s) Old World Record (s) Since her winning times were less than the old world records, or the old world records were greater than her winning times, we can describe her achievement in the form of an inequality. A mathematical inequality may contain a symbol such as <,, >,, or. In a first-degree inequality, such as x + 2 > 7, the variable has the exponent 1. To solve an inequality, find values of the variable that make the inequality true. For example, the inequality x + 2 > 7 is true for x = 5.1, x = 6, x = 7.25, and all other real values of x greater than 5. These values are said to satisfy the inequality. We write the solution as x > 5. INVESTIGATE & INQUIRE 1. Passenger aircraft land at 240 km/h, but their speeds on their landing approaches are higher than this. When passenger aircraft descend for a landing, their speeds during descent are given by the inequalities s and s , where s is the speed in kilometres per hour. a) Solve the equations s 320 = 0 and s 320 = 80. b) Solve the inequalities s and s using the same steps as you used in part a). 2. What is the lowest speed at which a passenger aircraft can descend? 3. What is the highest speed at which a passenger aircraft can descend? 4. a) List the three greatest whole-number solutions for s b) Use substitution to show that your three values from part a) satisfy both inequalities. 72 MHR Chapter 1

2 5. Solve each of the following inequalities using the same rules used to solve equations. a) 4x + 7 < 15 b) 7x + 2 > 23 1 c) x > 7.3 d) x 5 < a) The table shows the results of various operations on both sides of the inequality. Copy and complete the table by replacing each with > or <. Original Inequality Operation Add 3 Subtract 3 Multiply by 3 Multiply by 3 Divide by 3 Resulting Inequality ( 3) 6 ( 3) Divide by b) State the operations that reverse the direction of the inequality symbol. 7. Test your statement from question 6b) by determining the results of the following operations. Inequality Operation a) 4 > 3 Add 5 b) c) 3 < 1 Subtract 2 d) e) 2 > 1 Multiply by 4 f) g) 4 > 3 Multiply by 2 h) i) 3 < 6 Divide by 3 j) k) 4 < 2 Divide by 2 l) Inequality 2 < 6 1 > 4 3 < 2 4 < 3 2 > 2 4 > 8 Operation Add 1 Subtract 2 Multiply by 3 Multiply by 1 Divide by 2 Divide by 4 8. Use your answers from questions 6 and 7 to solve each of the following. Use substitution to verify your solutions. x a) 4x > 4 b) < 1 3 First-degree inequalities in one variable can be solved by performing the same operations on both sides to isolate the variable. When multiplying or dividing both sides of an inequality by a negative number, reverse the direction of the inequality symbol. 1.9 Solving First-Degree Inequalities MHR 73

3 In the following examples and problems, assume that all variables represent real numbers. EXAMPLE 1 Solving an Inequality Solve and check 3x 2 < 13. SOLUTION 3x 2 < 13 Add 2 to both sides: 3x < x < 15 Divide both sides by 3: 3x 15 < 3 3 x < 5 Check. Try x = 4: L.S. = 3x 2 R.S. = 13 = 3(4) 2 = 10 L.S. < R.S. The solution is any real number less than 5. EXAMPLE 2 Solving and Graphing Solve 2(3 x) 1 7. Graph the solution. SOLUTION 2(3 x) 1 7 Expand to remove brackets: 6 2x x 7 Subtract 5 from both sides: 5 2x x 2 2x 2 Divide both sides by 2: 2 2 x 1 The graph is as shown. The closed dot at x = 1 shows that 1 is included in the solution. When you multiply or divide by a negative number, reverse the direction of the symbol MHR Chapter 1

4 EXAMPLE 3 Solving an Inequality Involving Fractions 3x x Solve + >5. Graph the solution. 4 2 SOLUTION The LCD is 4. 3x 4 x + > x x x + 2x > 20 5x > 20 Divide both sides by 5: 5x 20 > 5 5 x > 4 Check. Try x = 8: 3x x L.S. = R.S. = 5 3(8) (8) = Multiply both sides by 4: + > 5 = = 10 L.S. > R.S. The solution is any real number greater than 4. The graph is as shown. The open dot at x = 4 shows that 4 is not included in the solution Solving First-Degree Inequalities MHR 75

5 EXAMPLE 4 Solving an Inequality Involving Decimals Solve 0.5(x + 4) 0.2(x + 6) 0.5(x + 1) 2.8. Graph the solution. SOLUTION 0.5(x + 4) 0.2(x + 6) 0.5(x + 1) 2.8 Expand to remove brackets: 0.5x x x Simplify: 0.3x x 2.3 Subtract 0.8 from both sides: 0.3x x x 0.5x 3.1 Subtract 0.5x from both sides: 0.3x 0.5x 0.5x x 0.2x 3.1 Divide both sides by 0.2: 0.2x x 15.5 The graph is as shown. Note that Example 4 could also be solved by isolating the variable on the right side. After expanding and simplifying, 0.3x x 2.3 Add 2.3 to both sides: 0.3x x Subtract 0.3x from both sides: x Divide both sides by 0.2: 15.5 x or x EXAMPLE 5 Selling Hiking Staffs Volunteers from a hiking association are selling hiking staffs as a fundraiser. The cost of making the staffs is a fixed overhead of $2000, plus $10 per staff. Each staff is sold for $30. What number of staffs must be sold for the revenue to exceed the cost? SOLUTION 1 Paper-and-Pencil Method Let x represent the number of staffs made and sold. The cost of making the staffs is C = x. The revenue from selling the staffs is R = 30x. 76 MHR Chapter 1

6 For the revenue to exceed the cost, R > C, so 30x > x. 30x > x Subtract 10x from both sides: 30x 10x > x 20x > x 2000 Divide both sides by 20: > x x > 100 Over 100 staffs must be sold for the revenue to exceed the cost. SOLUTION 2 Graphing-Calculator Method Let x represent the number of staffs made and sold. The cost of making the staffs is C = x. The revenue from selling the staffs is R = 30x. Enter the equations y = x and y = 30x in the Y= editor of a graphing calculator. Graph the equations using suitable values of the window variables. Use the intersect operation to find the coordinates of the point of intersection. For these graphs, the window variables include Xmin = 0, Xmax = 150, Ymin = 0, and Ymax = The value of x where the graphs intersect is 100. For the revenue to exceed the cost, R > C, so 30x > x. The graph of y = 30x is the graph that starts at the origin. This graph is above the graph of y = x when 30x > x. So, R > C when x > 100. Over 100 staffs must be sold for the revenue to exceed the cost. 1.9 Solving First-Degree Inequalities MHR 77

7 Key Concepts The results of performing operations on an inequality are summarized in the table. Similar results are observed for inequalities that include the symbols <,, and. These results and the methods used to solve equations can be used to solve inequalities. Original Inequality Operation Add c Subtract c Multiply by c, c > 0 Multiply by c, c < 0 Divide by c, c > 0 Divide by c, c < 0 Resulting Inequality a + c > b + c a c > b c ac > bc, c > 0 ac < bc, c < 0 a c a c b >, c > 0 c b <, c < 0 c For a graph of an inequality on a number line, a closed dot shows that an endpoint is included, and an open dot shows that an endpoint is not included. Communicate Your Understanding 1. Describe each step in solving 5x 7 > 2x + 11 algebraically. 2. Describe when it is necessary to reverse the direction of the inequality symbol when solving an inequality algebraically. 3. Describe how you would write two inequalities that have the solution x 3 and that have variables on both sides of the inequality symbol. 4. Describe how the graphs of x > 2 and x 2 compare. Practise A 1. Solve and check. a) y + 9 < 11 b) 2w + 5 > 3 c) 3x 4 5 d) 2z e) 3x < 6 f) 4t > 3t 4 g) 2(m 3) 0 h) 4(n + 2) 8 2. Solve and check. a) 2x + 1 > 2 b) 3x + 4 < 2 c) 6y + 4 5y + 3 d) 4z 3 3z + 2 e) 7 + 3x < 2x + 9 f) 5(2x 1) > 5 g) 2(3x 2) 4 h) 4(2x + 1) 2 3. Solve. Graph the solution. a) 6x + 2 4x + 8 b) 4x 1 > x + 5 c) 2(x + 3) < x + 4 d) 3(x 2) > x 4 e) 3(y + 2) 2(y + 1) f) 3(2z 1) 2(1 + z) g) 6x 3(x + 1) > x + 5 h) 2(x 2) 1 < 4(1 x) MHR Chapter 1

8 4. Solve. a) 6 2x > 4 b) 8 3x < 5 c) 3y 8 7y + 8 d) 6 3c 2(c 2) e) 4(1 x) 3(x 1) f) 2(3 + x) < 4(x 2) g) 4x 3(2x + 1) 4(x 3) h) 2(3t 1) 5t > 6(1 t) Solve. Graph the solution. a) y w + 2 < 1 b) + 2 > c) 2x 3z d) e) 1.2x 0.1 > 3.5 f) 0.8x < 2.3 g) q h) n Solve. a) 2(1.2a + 2.5) > 0.2 b) 4( x) 5.2 c) 0.75y 2.6 < 0.25y 3.1 d) 3(1.3n + 0.3) 3.5n e) 1.5(x + 2) + 1 > 2.5(1 x) 0.5 f) 2(1.5x + 1) 1 < 5(0.2x + 0.3) Solve. a) x + 1 x + 2 < 2 3 b) 2 x 2x c) z + 2 z 1 > d) 2 3x 2 + 3x Apply, Solve, Communicate 8. Art supplies Katrina has a $50 gift voucher for an arts supply store. She wants to buy a sketch pad and some markers. Including taxes, a sketch pad costs $18 and a marker costs $4. Use the inequality 4m to determine the number of markers, m, she can buy. 9. Measurement In ABC, A is obtuse and measures (5x + 10). Solve the inequalities 5x + 10 > 90 and 5x + 10 < 180 to find the possible values of x. B 10. Application The cost of an extra large tomato and cheese pizza is $12.25, plus $1.55 for each extra topping. a) Let n represent the number of extra toppings. Write an expression, including n, to represent the total cost of the pizza. b) Suppose you have $20 you can spend on the pizza. Write and solve an inequality to find the number of extra toppings you can afford. 11. Geometry ABC is not an acute triangle. B is the greatest angle. The measure of B is (4x). What are the possible values of x? A (4x) 1.9 Solving First-Degree Inequalities MHR 79 B C

9 12. Weekly earnings Mario earns $15/h after taxes and other deductions. He spends a total of $75/week on lunches and travel to and from work. a) Write an expression to represent how much Mario has at the end of a week in which he works t hours. b) Write and solve an inequality to determine how many hours Mario must work to have at least $450 at the end of the week. 13. Baseball caps A college baseball team raises money by selling baseball caps. The cost of making the caps includes a fixed cost of $500, plus $7 per cap. The caps sell for $15 each. What is the minimum number of caps the team can order in one batch and still raise money? 14. Populations Paris and Aylmer are towns in Ontario. From 1991 to 1996, the population of Paris increased from 8600 to Over the same period, the population of Aylmer increased from 6200 to If each population continues to increase at the same rate as it did from 1991 to 1996, over what time period would you expect the population of Aylmer to be greater than the population of Paris? 15. Measurement a) What values of x give this rectangle a perimeter of more than 32 cm? b) What values of x give the rectangle an area of less than 40 cm 2? c) Communication In part b), does x have a minimum value? Explain. 16. Modelling problems algebraically Determine the values of x that give this triangle a perimeter of no more than 15 and no less than Solve 3(x + 2) 5 2(1 x) + 4. Graph the solution. 18. Charity auction A charity wants to raise at least $ for hospital equipment. A car dealer donates a car to be used as a raffle prize. The charity determines that it can sell 1500 to 2000 tickets. If advertising and other costs are estimated to be $4500, a) what should the charity set as the ticket price? b) what is the range of profit expected? 2 cm (3x 7) cm 2(x 1) x 3x 1 80 MHR Chapter 1

10 19. Inquiry/Problem Solving Write the following inequalities. Have a classmate solve them. a) variables on both sides and the solution x 2 b) brackets on both sides and the solution x > 3 c) denominators of 3 and 2 and the solution x < 0 C 20. Driving times Jason left Hamilton at 10:00 and drove 620 km to Montréal at an average speed of 80 km/h. Hakim left Hamilton an hour later and drove to Montréal at an average speed of 100 km/h. Between what times of the day was Hakim further from Hamilton than Jason was? 21. a) Try to solve the equation 4x + 2(x + 1) = 6x 2. What is the result? b) What real values of x satisfy the equation? c) Try to solve the inequality 4x + 2(x + 1) > 6x 2. What is the result? d) What real values of x satisfy the inequality? 22. Technology a) Predict the graph of the following expression, if it is drawn using a graphing calculator. y = (x 3)(x < 2) Check your prediction using a graphing calculator in the dot mode. Describe the result. b) Repeat part a) for the expression y = (2 x)(x < 5). c) Repeat part a) for the expression y = (x + 4)(x > 4). A CHIEVEMENT Check Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application For what values of x is the triangle possible? 2x + 3 x + 7 3x Solving First-Degree Inequalities MHR 81