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Consider any of these inequalities:
A B ,
A B ,
A > B , A <
B. |
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For each such inequality,
it's ASSOCIATED EQUATION is A = B. |
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We will abbreviate the phrase
"ASSOCIATED EQUATION" to
"AE". |
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The AE for the inequality "3x - 2y 5"
is "3x - 2y = 5" . |
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Let A and B be algebraic expressions
involving the variables x, y, or others.
The graph of the equation
A = B divides up space into pieces; |
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within each piece, |
either [1] ALL points satisfy A > B , or
or [2] ALL points satisfy A < B. |
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In our situations, A and B will usually involve the
two variables "x" and "y", the AE's will have
straight-line graphs, and therefore the pieces into
which A = B divides space will be "half-planes",
that is, the set of points on one side of a
straight line. |
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To graph an inequality
(such as A B), first
graph it's AE. Then test each piece into which
that graph divides space (with "TEST POINTS":
red points in figure
below). Choose or mark those pieces whose test
points satisfy the given inequality (in our
example, A B). Thus, in
the examples below, the origin ( 0 , 0 ) makes a
good test point, for which the stated inequality
" x 2 + y 2 1 "
is TRUE. |
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MARKING YOUR GRAPH
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ARROWS are used when marking several overlapping regions. Thus,
the graph at the right marks the interior of the circle with
center (0,0) and radius 1, which solves the inequality
" x2 + y2 1 ". |
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The test points are C = (0,0) and
D = (2,0): C
satisfies " x2 + y21"
since
" 02 + 021".
On the other hand, D fails to satisfy
" x2 + y2 1 ", because
" 22 + 02 = 4 > 1 ". Thus, using Fact 3
above, the solution set for our inequality is the set of
points on the same side of the circle as is
C. |
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