160 Chapter 2 Linear Equations and Inequalities in One Variable Since Blair and Tatiana are traveling toward each other, their combined distance traveled is 390 mi. We use this to write the equation. Tatiana’s distance + Blair’s distance = 390 60t + 70t = 390 130t = 390 130t 130 = 390 130 Express the relationship. Combine like terms. Divide each side by 130. t = 3 Simplify. So, they both traveled for 3 hr until they met. Note that Tatiana drove 60(3) = 180 miles in 3 hr and Blair drove 70(3) = 210 miles in 3 hr. 4b. Highway inspectors are examining a highway. Two inspectors start from the same point but one travels north and the other travels south. The inspector heading north leaves at 8:00 a.m. and travels at an average speed of 40 mph. The inspector heading south leaves at 10:00 a.m. the same day and travels at an average speed of 50 mph. When will the inspectors finish examining a 260-mi portion of the highway? Solution 4b. What is unknown? The time that each inspector travels is unknown. The inspector heading south leaves 2 hr after the inspector heading north. Let t = time for the inspector heading north and t - 2 = time for the inspector heading south. What is known? The total distance covered is 260 mi. The north inspector has a rate of 40 mph and the south inspector has a rate of 50 mph. Rate × Time = Distance North 40 t 40t South 50 t - 2 50(t - 2) Since their total distance is 260 mi, the equation is as follows. 40t + 50(t - 2) = 260 40t + 50t - 100 = 260 90t - 100 = 260 90t - 100 + 100 = 260 + 100 90t = 360 90t 90 = 360 90 t = 4 Express the relationship. Apply the distributive property. Combine like terms. Add 100 to each side. Simplify. Divide each side by 90. Simplify. The inspector heading north will be done in 4 hr and the inspector heading south will be done in 4 - 2 = 2 hr. So, the job will be complete at 12:00 p.m. Student Check 4 Use the distance formula to write an equation that will solve each problem. Solve the equation and answer the question using a complete sentence. a. Two cars leave the same town heading in opposite directions. One car travels at 50 mph and the other at 70 mph. How long will it take for the cars to be 480 mi apart? b. Tom and Barbara live 500 mi apart. They leave their homes and travel toward one another. Tom travels at an average speed of 65 mph and Barbara travels at an average speed of 60 mph. Tom leaves 1 hr before Barbara. How long will they each travel before they meet?
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