Sample Problem 3

I have been told that I can solve any equation involving absolute value simply by squaring both sides, and solving the resulting quadratics.

Is this always true?

* Level 1: General information

The absolute value of an integer, such as 4 or -5, returns only the SIZE of the number, not its direction (+ or -). This means that it always returns a positive result.

But what about VARIABLES? What do you understand by the "absolute value of x"?

* Level 2: Specific information

The absolute value of any variable x may be thought of as its DISTANCE FROM ZERO. An equation such as | x - 2 | = 5, then, refers to those values of x which lie exactly 5 units away from 2 (i.e. -3 and 7).

* Level 3: Suggest action

Since | x | = sqrt(x² ) then squaring both sides of the equation | x - 2 | = 5

--> ( x - 2 )² = 25

--> x² - 4x + 4 = 25 (expand)

--> x² - 4x - 21 = 0 (subtract 25 from both sides)

--> (x + 3)(x - 7) = 0 (factorise)

This certainly produces the same solutions as the original equation.

But will it always work?

* Level 4: Suggest sequence

Consider the GRAPHS and TABLE OF VALUES of both

y = | x | and y = x² .

Both are "even" functions, symmetrical about the y-axis. What about the graphs of y = | x - 2 | and y = (x - 2)² ?

* Level 5: Demonstrate

Try graphing both sides of the equations | x - 2 | = 5 and ( x - 2 )² = 25. Are you convinced now? Is there any way to PROVE that this method will always work, other than to state the rule that | x |² = x² for all values of x?

This problem requires a different type of support to the others, owing largely to its open-ended nature. There is no definitive answer in such a case, and many approaches and paths are possible. The help available should not be too prescriptive; it should, instead, actually pose questions and suggest alternative ideas and approaches, encouraging the learner to broaden the problem focus rather than to narrow it.

Sample Problem 2. Conclusion.

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Last updated: 1st May, 1996
Stephen Arnold
crsma@cc.newcastle.edu.au
© 1996 The University of Newcastle

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