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Math 141 HW3C

        Please follow the usual directions for written homework. Use only the material from Sections 1.9, 2.1-2.4.

Letter to a Friend 1 (5 points). The exit numbers on I-5 in the state of Washington measure the distance from the Washington-Oregon border. So for example, the Shoreline exit, Exit 176, is 176 miles from the border.

        Once you cross into Oregon, the exit numbers mesaure the distance to the California border. But for this problem, let’s change the exit numbers in Oregon to measure the signed distance from the Washington-Oregon border, where the positive direction points north. So all the exit numbers in Oregon are now negative, and Exit −12 for example, is 12 miles south of the WA-OR border.

        In other words, we are thinking of I-5 as a straight number line (ignore the curves in the road), call it the x-axis, with x = 0 at the border and the positive direction pointing due north. And we’ll assume that the road runs forever in both directions. So the variable x gives a position on I-5, measured as the signed distance from the WA-OR border.

        The purpose of this letter is to explain the meaning of the some value functions and to explain both logically and mathematically how to solve some absolute value equalities and inequalities in the context of this situation.

        In your letter, make up your own problems, incorporating the following types of questions. Be sure to draw a picture of I-5 (ie. a number line) for each to help with your explanations.

        Here are the types of problems you should include. You should make up some of your own as well. Except for part (a), do not use these exact prob-lems. You do not need to stick with this order. Rather, order the questions in a way that makes most sense to you.

        (a) Start by explaining the meaning of the function f(x) = |x| in the context of the I-5 number line. What is its input? Its output? Give concrete examples by evaluating the function for some inputs of your choosing

        (b) Explain the meaning of the function g(x) = |x − 176|. What is its input? Its output? How about the function h(x) = |x + 132|? Give concrete examples by evaluating the function for some inputs of your choosing.

        (c) Find a function that measures the distance from Exit 23. Find another that measures the distance from Exit −189.

        (d) Write an absolute value inequality that describes locations on I-5 at most 87 miles from Exit 176. Then solve the inequality. Describe these positions. Sketch the solution set on your number line. Write the solution set using set notation, not interval notation.

        (e) What does it mean (in this context) if you are asked to solve the equation |x − 89| = 55? Solve the equality using common sense. Then solve it mathematically. Be sure to include a graph and the solution set.

        (f ) What does it mean if you are asked to solve the inequality |x + 154| > 55. Solve the inequality using common sense. Then solve it mathematically. Be sure to include a graph and the solution set.

        (g) Write an equality that describes the locations on I-5 that are at least 54 miles from Exit −123. Then solve the inequality and describe all such locations.

        (h) What locations on I-5 are exactly three times as far from Exit 17 as from Exit −111? Turn this question into an absolute value equality. Solve it. Then describe all such locations.

Reflection (5 points). Read the solution (from a student) to problem 2a of HW2C in the file HW2CS.pdf in the Homework module of our Homepage. Then address the following questions.

i) What does the letter do particularly well?

ii) What might you have done better in your solution?

iii) There is a small error in the solution. Identify and correct it.