182 Chapter 2 Linear Equations and Inequalities 51. How many quarts of 85% chlorine solution must be mixed with 5 quarts of 25% chlorine solution to obtain a 45% chlorine solution? 2.5 quarts of 85% 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? 17.5 L of 58% 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.) 20 L of water 54. How much pure water must be mixed with 10 oz of a 60% alcohol solution to obtain a 25% alcohol solution? 14 oz of water 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. The Japanese bullet train travels 300 km/hr and the Acela Express travels 240 km/hr. 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 .) 150 mph Section 2.8 Linear Inequalities 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. The following inequalities are linear equalities in one variable. 2x 3 6 11 4z 3 7 0 a 4 5.2y 10.4 The number line is a useful tool to visualize the solution set of an equation or inequality. For example, the solution set to the equation x 2 is {2} and may be graphed as a single point on the number line. x 2 6 5 4 3 2 1 0 1 2 3 4 5 6 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 Writing Translating Expression Geometry Scientific Calculator Video
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