comparisons and operations
Algebraic operations on comparisons
When you studied linear comparisons, one way to
"solve" such a comparison (that is, to find its solutions) was
to perform algebraic operations on both sides of the comparison in order
to obtain an equivalent comparison: one with the same
solutions. By performing a series of such operations,
it is possible to obtain a comparison equivalent to
the original, and with solutions that are easy to identify.
The objective of this task is to investigate algebraic operations on
linear and quadratic comparisons.
The central question is: Which operations are "allowed"? Which operations
generate an equivalent comparison, and under what circumstances?
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The
Comparisons and algebraic operations
tool represents a comparison symbolically and graphically, and allows
you to change it by performing algebraic operations on one
side of the comparison or on both sides simultaneously.
To explore the effect of algebraic operations, construct
linear or quadratic comparisons and perform algebraic operations on
one or both sides:
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Adding a function (constant, linear, quadratic...).
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Subtracting a function.
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Multiplying by a function.
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Dividing by a function.
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Prepare a report on the effect of algebraic operations on the solutions of comparisons
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Report on various operations and their
effect on the solutions of comparisons. Do the solutions change?
If so, how?
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Explain how the effect of algebraic operations on the solutions of a comparison
depends on:
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The type of comparison (equation, inequality).
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The type of operation (addition, multiplication, etc.).
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The function applied to the side/s of the comparison.
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Other factors.
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Suggest generalizations. Provide examples and explanations
for your generalizations.
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Report on interesting special cases.
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Report on conjectures and questions that arise.
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