Section 2.8 Linear Inequalities 165 47. A boat in distress, 21 nautical miles from a marina, travels toward the marina at 3 knots (nautical miles per hour). A coast guard cruiser leaves the marina and travels toward the boat at 25 knots. How long will it take for the boats to reach each other? 48. An air traffic controller observes a plane heading from New York to San Francisco traveling at 450 mph. At the same time, another plane leaves San Francisco and travels 500 mph to New York. If the distance between the airports is 2850 mi, how long will it take for the planes to pass each other? 49. Surfer Sam purchased a total of 21 items at the surf shop. He bought wax for $3.00 per package and sunscreen for $8.00 per bottle. He spent a total amount of $88.00. How many of each item did he purchase? 50. Tonya Toast loves jam. She purchased 30 jars of gourmet jam for $178.50. She bought raspberry jam for $6.25 per jar and strawberry jam for $5.50 per jar. How many jars of each did she purchase? 51. How many quarts of 85% chlorine solution must be mixed with 5 quarts of 25% chlorine solution to obtain a 45% chlorine solution? 52. How many liters of a 58% sugar solution must be added to 14 L of a 40% sugar solution to obtain a 50% sugar solution? Expanding Your Skills 53. How much pure water must be mixed with 12 L of a 40% alcohol solution to obtain a 15% alcohol solution? (Hint: Pure water is 0% alcohol.) 54. How much pure water must be mixed with 10 oz of a 60% alcohol solution to obtain a 25% alcohol solution? 55. Amtrak Acela Express is a high-speed train that runs in the United States between Washington, D.C. and Boston. In Japan, a bullet train along the Sanyo line operates at an average speed of 60 km/hr faster than the Amtrak Acela Express. It takes the Japanese bullet train 2.7 hr to travel the same distance as the Acela Express can travel in 3.375 hr. Find the speed of each train. 56. Amtrak Acela Express is a high-speed train along 16 the northeast corridor between Washington,D.C. and Boston. Since its debut, it cuts the travel time from 4 hr 10 min to 3 hr 20 min. On average, if the Acela Express is 30 mph faster than the old train, find the speed of the Acela Express. (Hint: 4 hr 10 min 4 hr .) Linear Inequalities Section 2.8 1. Graphing Linear Inequalities Consider the following two statements. 2x 7 11 and 2x 7 6 11 The first statement is an equation (it has an sign). The second statement is an inequality (it has an inequality symbol, ). In this section, we will learn how to solve linear inequalities, such as 2x 7 11. A Linear Inequality in One Variable A linear inequality in one variable, x, is any inequality that can be written in the form: ax b 6 c, ax b c, ax b 7 c, or ax b c, where a 0. Concepts 1. Graphing Linear Inequalities 2. Set-Builder Notation and Interval Notation 3. Addition and Subtraction Properties of Inequality 4. Multiplication and Division Properties of Inequality 5. Inequalities of the Form a x b 6. Applications of Linear Inequalities
miller_beginning_intermediate_algebra_4e_ch1_3
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