  Assignment 13

 "It is hard to convince high school students that they will encounter a lot of problems more difficult than those of algebra and geometry." -- (Edward G. Howe) Mathematical Drought in Western World (500 - 1200): There was little motivation to study the physical world in western civilization, as Europeans were trapped in primitive fundamentalism. The marvelous cathedrals and monasteries constructed at Chartes, Orleans, Rheims, Canterbury, and other locations were indeed accomplishments that required engineering and mathematical skills, but in general, the Catholic Church preached that the Christian God ruled the universe and that man's role was to serve Him. However, Arabs and Hindus (relatively isolated, and shunned by the West) were making advancements in math and science. They set up libraries, observatories, and research institutes. Why was Herkimer upset when his vacationing girl friend said that she hoped to see him pretty soon? Answer: He thought he was pretty now. Things Herky would like to know: Why do banks charge you a non-sufficient funds fee on money that they already know you don't have? Why are there five syllables in the word monosyllabic ? Reading: Section 2.6, pages 108-111. Written: Pages 111-112/24-26, then 27-45 (odds). Do the graphs neatly. No slop, please! Mathematical word analysis:RADIUS: From the Latin word radius , meaning "the spoke of a wheel." (The term was first used in 1569 by French mathematician Peter Ramus.)

OK, good people, this section should make total sense. Don't be mechanical. Think about what you are doing when you graph linear inequalities.

Here is just a simple example of the thinking: Suppose you want to graph the inequality

4x + 2y < 8

This is a conditional statement. That is, it has no truth value until you put in x- and y-values. If I substitute x = 3 and y = 8, the statement is false, since 12 + 16 is not less than 8. In other words, the point (3,8) does not satisfy the relation. (The collection of points described the the inequality is a relation, not a function.) The point (3,8) is not on the graph of the relation 4x + 2y < 8. If I substitute (1,1), the statement is true, since 4 + 2 is less than 8. OK, this means (1,1) is on the graph of the relation. 4x + 2y < 8.

Let's now solve the inequality for y. What follows should make sense, and you should be able to read it.

4x + 2y < 8 ==> 2x + y < 4 ==> y < -2x + 4.

So, if the y-value of a point is less than -2 times the x value + 4, the x-and y-value of the point make the conditional statement true.

The graphing procedure is relatively simple. Graph the line y = -2x + 4. Note that points on this line do not satisfy the inequality, so the line should not be solid. You should convince yourself that the set of points that satisfy the given inequality lie below this line. Don't just accept this. Make sure it makes sense to you.

MATH POWER TO ALL.

 Problem: Graph the inequality 8x + 2y È 32. Solution (with communication): 8x + 2y È 32 ==> 2y È -8x + 32 ==> y È -4x + 16. To produce a graph of the given inequality, first graph the solid line y = -4x + 16. Then, shade in all of the region (a half-plane) below this line. (The line is part of the collection of points in the solution set, so it should appear in solid line, not a dashed line.) Problem: Graph the inequality 3x - y < 15. Solution (with communication): 3x - y < 15 ==> -y < -3x + 15 ==> y > 3x - 15. To produce a graph of the given inequality, first graph the dashed line y = 3x - 15. Then, shade in all of the region (a half-plane) above this line. (The line is not part of the collection of points in the solution set, so it should be a dashed line, not a solid line.)