A76 APPENDIX Review 57. Constructing a Box An open box is to be constructed from a square piece of sheet metal by removing a square of side 1 foot from each corner and turning up the edges. If the box is to hold 4 cubic feet, what should be the dimension of the sheet metal? 58. Constructing a Box Rework Problem 57 if the piece of sheet metal is a rectangle whose length is twice its width. Explaining Concepts 59. Critical Thinking Make up a word problem that requires solving a linear equation as part of its solution. Exchange problems with a friend. Write a critique of your friend’s problem. 60. Critical Thinking You are the manager of a clothing store and have just purchased 100 dress shirts for $20.00 each. After 1 month of selling the shirts at the regular price, you plan to have a sale giving 40% off the original selling price. However, you still want to make a profit of $4 on each shirt at the sale price. What should you price the shirts at initially to ensure this? If, instead of 40% off at the sale, you give 50% off, by how much is your profit reduced? 61. Computing Average Speed In going from Chicago to Atlanta, a car averages 45 miles per hour, and in going from Atlanta to Miami, it averages 55 miles per hour. If Atlanta is halfway between Chicago and Miami, what is the average speed from Chicago to Miami? Discuss an intuitive solution. Write a paragraph defending your intuitive solution. Then solve the problem algebraically. Is your intuitive solution the same as the algebraic one? If not, find the flaw. 62. Speed of a Plane On a recent flight from Phoenix to Kansas City, a distance of 919 nautical miles, the plane arrived 20 minutes early. On leaving the aircraft, I asked the captain, “What was our tail wind?” He replied,“I don’t know, but our ground speed was 550 knots.” Has enough information been provided for you to find the tail wind? If possible, find the tail wind. (1 knot 1 nautical mile per hour) = 63. Critical Thinking Without solving, explain what is wrong with the following mixture problem: How many liters of 25% ethanol should be added to 20 liters of 48% ethanol to obtain a solution of 58% ethanol? Now go through an algebraic solution. What happens? OBJECTIVES 1 Use Interval Notation (p. A77) 2 Use Properties of Inequalities (p. A78) 3 Solve Inequalities (p. A80) 4 Solve Combined Inequalities (p. A81) 5 Solve Inequalities Involving Absolute Value (p. A82) Suppose that a and b are two real numbers and a b. < The notation a x b < < means that x is a number between a and b . The expression a x b < < is equivalent to the two inequalities a x < and x b. < Similarly, the expression a x b ≤ ≤ is equivalent to the two inequalities a x ≤ and x b. ≤ The remaining two possibilities, a x b ≤ < and a x b, < ≤ are defined similarly. Although it is acceptable to write x 3 2, ≥ ≥ it is preferable to reverse the inequality symbols and write instead x 2 3 ≤ ≤ so that the values go from smaller to larger, reading from left to right. A statement such as x 2 1 ≤ ≤ is false because there is no number x for which x 2 ≤ and x 1. ≤ Finally, never mix inequality symbols, as in x 2 3. ≤ ≥ A.9 Interval Notation; Solving Inequalities Now Work the ‘Are You Prepared?’ problems on page A84. • Algebra Essentials (Section A.1, pp. A1–A10) PREPARING FOR THIS SECTION Before getting started, review the following:
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