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Rational
Inequalities |
The
graph of the translated equilateral (or rectangular) hyperbola
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The graph of the
translated equilateral (or rectangular) hyperbola
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The graph of the given rational function is translated equilateral (or rectangular)
hyperbola.
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rational function of the
form
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can be rewritten into
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where |
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the vertical asymptote,
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the horizontal
asymptote |
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and the
parameter
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Therefore, values of the vertical and
the horizontal asymptote correspond to the coordinates of the horizontal and the vertical translation
of the source
equilateral hyperbola y
= k/x, respectively.
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where, a
= 1, b
=
-2
and c
=
1,
d
=
1 |
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has
the vertical asymptote |
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the
horizontal asymptote |
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and
the parameter |
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its source function is the equilateral or rectangular
hyperbola |
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The graph of given
rational function is shown below.
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| Example:
Find the solutions
of the inequality
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Solution:
The solution set of the given rational inequality includes all numbers
x
which make the inequality less then or equal to 0, or which make the sign of the rational expression
to be negative or 0.
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A rational expression is negative if the numerator and the denominator
have different signs, and the rational expression equals 0 when
its numerator is equal to 0 that is,
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therefore, we have to solve two simultaneous
inequalities.
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We graph the numerator and the
denominator in the same coordinate system to find all points of
the x-axis
that satisfy given inequality.
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The zero points of the numerator and
the denominator divide the x-axis
into four intervals at which given rational expression changes
sign.
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| x2
+ 2x -
3 = 0, a = 1, b = 2 and
c = -3
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The solutions of the two pairs of the simultaneous
inequalities are intersections of sets of their partial solutions,
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as is shown below.
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Therefore, the solution set is (-
oo,
- 3]
U
(-
2,
1].
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| Intermediate
algebra contents |
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