The Procedural Approach (PA) lecture slides - - PDF document

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The Procedural Approach (PA) lecture slides - - PDF document

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The Procedural Approach (PA) lecture slides

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  • We start by the idea of using graphing to

compare functions

  • Next, we introduce the concept of absolute

value

  • And, finally, we learn how to solve inequalities

with absolute value

  • The Visual Approach lecture slides

VA-1 VA-2

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SLIDE 13

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  • Comparing the values of two functions, by looking at their graphs

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VA-3 VA-4

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SLIDE 14

Jane and Joe are measuring the circumference of a dime with a string.

A LESSON ON ABSOLUTE VALUE

Tom knows the true length of the circumference: 56 mm. He calculates the difference between the true length and the measurements: 56 - 55 = 1 56 - 58 = -2 He says: Since 1 > -2 then Jane made a bigger mistake than Joe. Jane' s result is: 55 mm Joe's result is: 58 mm

Do you agree with Tom?

Sometimes we are not interested in knowing whether a measurement was less than or greater than the true value but only in the MAGNITUDE of the difference. We call this magnitude THE ABSOLUTE VALUE of the number obtained as the difference.

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|2 - 1 | < |2 - 4| The absolute value of the difference 2 - 1 is equal to 1. |2 - 4 | = |-2 | = 2 2 = the opposite of -2 = -(-2) The absolute value of the difference 2 - 4 is equal to 2. |2 - 1 | = | 1 | = 1

VA-5 VA-6

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SLIDE 15

In general, we define the absolute value of a number x as equal to x for positive numbers x, and equal to the opposite of x for negative numbers x. As for the number zero, we decide that |0| = 0. The absolute value of a number is thus never negative. We can think of the absolute value of a number as a function, defined as follows:

|x| =

x if x is positive or 0

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if x is negative

Here is a visual representation of this definition, using the number line:

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1 2 3 4 5 |3|=3 |-2| = -(-2) = 2

The absolute value of a positive number is equal to itself. The absolute value of a negative number is equal to its opposite. Graphing absolute value functions

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VA-7 VA-8

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SLIDE 16

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Exercise: Find all values of x for which |x - 1| < |x + 2|

/ / /)/ + ,.+ 1%-2

Suppose it is not so easy to see from the graph for what values of x the values of one function are less than the values of the other one. What does one do in such cases? We can use reasoning and algebra. Let's look again at the inequality | x - 1 | < |x + 2|. (a) |x - 1 | = x - 1 for x - 1 ≥ 0 (b) |x - 1| = -(x - 1) for x - 1 < 0 (c) |x + 2| = x + 2 for x + 2 ≥ 0 (d) |x + 2| = -(x + 2) for x + 2 < 0 So we have 4 cases to consider:

(a) & (c): for x ≥ 1 & x ≥ -2, the inequality is x – 1 < x + 2 (a) & (d): for x ≥ 1 & x < -2, the inequality is x – 1 < -(x + 2) (b) & (c): for x < 1 & x ≥ -2, the inequality is –(x – 1) < x + 2 (b) & (d): for x < 1 & x < -2, the inequality is –(x – 1) < -(x + 2)

VA-9 VA-10

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SLIDE 17

Solving the inequality | x - 1 | < |x + 2|, continued

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(a) & (c): for x ≥ 1 & x ≥ -2, the inequality is x – 1 < x + 2 (a) & (d): for x ≥ 1 & x < -2, the inequality is x – 1 < -(x + 2) (b) & (c): for x < 1 & x ≥ -2, the inequality is –(x – 1) < x + 2 (b) & (d): for x < 1 & x < -2, the inequality is –(x – 1) < - (x + 2) Simplify (a) & (c): for x ≥ 1, the inequality is – 1 < 2 (a) & (d) contradiction (b) & (c): for x < 1 & x ≥ -2, the inequality is x > -1/2 (b) & (d) contradiction x > -1/2

Exercises

  • 1. Calculate:

||16-24| - |7-56|| In each of the exercises below, find the values of the number x for which the given inequality is true

  • 2. |x -1 | < |x + 1|
  • 3. |x + 3| < -3|x - 1|
  • 4. |2x - 1| < 5
  • 5. |2x - 1| > 5
  • 6. |50x - 1| < |x + 100|

Thank you for your attention!

VA-11 VA-12