  Assignment 5

 "The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit." -- (Alfred North Whitehead, 1861-1947) Eratosthenes (276 - 194 B.C.): This Greek mathematician calculated the circumference of the earth with an error of less than 2%. The computational process and reasoning used by Eratosthenes involved simple geometric concepts taught in modern geometry classes. Eratosthenes was the chief librarian of the library at Alexandria. When the library was destroyed in the fourth century, A.D., it had a collection of over 500,000 Greek works on thousands of topics, making it a center for research and learning in the Egyptian and Greek worlds. Why did Herkimer think that Cleopatra was a very negative person? Answer: Because he heard that she was the queen of denial. Herky's friends: TRUDY TULIPS...she liked to stroll through flower gardens.. SIR CUMFERENCE ...a man of royalty who studied round objects. Reading: Section 1.6, pages 41-44. Written: Pages 45-46/25-47 (odds). Mathematical word analysis:ADDITION: From the Latin word addere , which means "to put together." When we add numbers, we put them together to find what we call the sum.

When working with algebraic inequalities, we don't have balance in the sense of equality. Inequalities involve number relations such as "greater than" or "less than." When solving inequalities correctly, we frequently do a transformation that either preserves the indicated number relation, or a transformation that reverses it.

Here is a correct statement involving algebraic inequality:

x < y

==> -x > -y.

This statement, which involves multiplying both members of the inequality by -1, and reversing the relation <, should make sense. For instance, the number scale below shows two numbers, x and y, that satisfy the inequality x < y.

 X Y -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Now, let's put -x and -y in the appropriate places on this number line.

 -Y -X X Y -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Note that the number -x (read "the opposite of x") is greater than -y. In other words, the original algebraic statement, where < was reversed, makes sense.

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OK, here's problem #36 on page 45. Instructions say to solve the indicated inequality. Note the communication in the solution process.

5 - 5x > 4(3 - x)

==> 5 - 5x > 12 - 4x

==> 5 - x > 12

==> -x > 7

==> x < -7.

In other words, any number that makes the given conditional statement true must have the property that it is less than -7. In the solution process, the relation > was preserved until the final step, when the transformation required that it be reversed.

Mathematics is a powerful language. Use it properly, and use it efficiently.

 Problem: Solve (1/2)x- 4 > -6. Solution (with communication): (1/2)x- 4 > -6 ==> 2[(1/2)x- 4] > 2(-6) ==> x - 8 > -12 ==> x > -4. Conclusion: Any number greater than -4 will make the conditional statement true. Problem: Solve -8 < (2/3)x - 4 < 10. Solution (with communication): -8 < (2/3)x - 4 < 10 ==> -4 < (2/3)x < 14 ==> -12 < 2x < 42 ==> -6 < x < 21. Conclusion: Any number between -6 and 21 will make the conditional statement true. Problem: Solve: 3x + 2 < -10 or 4 - 2x < -4. Solution (with communication): 3x + 2 < -10 or 4 - 2x < -8 ==> 3x < -12 or -2x < -12 ==> x < -4 or x > 6. Conclusion: The compound conditional statement is true for any number that is less than -4 or greater than 6. Problem: I have less than \$2.00 in my pocket. I have 4 quarters, 5 dimes, and 3 pennies. What is the maximum number of nickels I can have? Solution (with communication): Let N = the number of nickels I have in my pocket. Premise ==> 4(25) + 5(10) + 3 + 5N < 200 ==> 153 + 5N < 200 ==> 5N < 47 ==> N < 9.4. Conclusion: I have at most 9 nickels in my pocket.