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| ALGEBRA
- solved problems |
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Rational
Inequalities
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| Solve the rational
inequality
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and
draw the graph of the rational function. |
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Solution:
The solution set of the given rational inequality includes all numbers
x
which make the inequality greater then or equal to 0, or which make the sign of the rational expression
to be positive or 0.
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A rational expression is positive if both the numerator and the denominator are positive
or if both are negative, 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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The solutions represented on the
number line are shown below.
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Thus, the solution set of the given
inequality written in the interval notation is (-
oo, -1)
U
[2, oo
).
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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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| Thus, given rational function |
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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, (see the above example of the rational inequality).
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Solve the rational
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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| Absolute value
inequalities |
| Solving
linear inequalities with absolute value |
| Graphical
interpretation of the
definition of the
absolute value of a function y =
f(x) will help us solve
linear inequality with absolute value. |
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| 96. |
Solve the absolute value
inequality | x
-
2
| < 3.
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| Solution:
Graphical
interpretation of the given inequality will help us find
values of x
for which left side of the inequality is less then or equal to
3. |
| For
values of x
for which y
is nonnegative, the graph of | y
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same as that of y =
x -
2.
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| For values of
x
for which y
is
negative, the graph of | y
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the graph of y
across the x
axis. |
| Since
the graph of y =
x -
2 has y
negative on
the interval (-
oo, 2) it is this part of the graph
that has to be reflected on the x
axis. |
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| The
graph shows that
values of x from
the closed interval [-1,
5] satisfy the given inequality. |
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same result can be obtained algebraically by solving the
compound inequality. |
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| x
Î [-1,
5] |
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| 97. |
Solve the absolute value
inequality | x
-
2
| > 3.
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| Solution:
Again, graphical
interpretation of the given inequality will help us find
values of x
for which left side of the inequality is greater than 3. |
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graph shows that
values of x from
the open intervals, (- oo, -1)
or (5,
oo)
satisfy the given inequality. |
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| The
same result can be obtained algebraically by solving the two
inequalities |
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