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ALGEBRA
- solved problems |
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Linear
inequalities
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Solving linear
inequalities |
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Solve given linear inequality.
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Solution: |
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x -
8
< 2x
-
4 |
- x
<
4
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·
(-1) |
x
>
-4 |
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The
solution of the inequality is every real number greater
than -4. |
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87. |
Solve inequality
3(x -
2) >
-2(1-
x).
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Solution:
3x -
6
>
-2
+
2x |
x > 4 |
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interval
notation
(4, oo) |
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The
open interval (4, oo)
contains all real numbers between given endpoints, where round
parentheses indicate exclusion of endpoints. |
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88. |
Solve given linear inequality.
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Solution:
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-4x
+
9 -
3x
<
6 - 5
+
5x |
-12x
<
-8 |
x
>
2/3 |
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interval
notation |
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The
half-closed (or half-open) interval
contains all real numbers between given endpoints, where the
square bracket indicates inclusion of the endpoint 2/3 and round
parenthesis indicates exclusion of infinity. |
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89. |
Solve inequality
(x
- 3)
· (x
+ 2)
> 0
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Solution:
The factor
x
- 3
has the zero at x
= 3, is negative for x
< 3
and is positive for x
> 3,
and |
the factor
x
+ 2
has the zero at x
= -2,
is negative for x
< - 2
and is positive for x
> -2, |
as
is shown in the table |
x |
-
oo |
increases |
-2 |
increases |
3 |
increases |
+
oo |
x
+ 2 |
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- |
0 |
+ |
+ |
+ |
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x
- 3 |
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- |
- |
- |
0 |
+ |
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(x
- 3)(x
+ 2) |
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+ |
0 |
- |
0 |
+ |
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Thus,
the given inequality is satisfied for -
oo
< x
< - 2
or 3
< x
< + oo |
in the interval notation (-
oo
, -2]
U [3,
+ oo
) |
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Solving
compound (double) inequalities
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90. |
Solve inequality
-4
< 2(x
- 3)
< 5
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Solution:
We want the x alone as middle term and only constants in the two outer terms. Remember,
while simplifying given compound inequality, the operations that we apply to a middle term we
should also do to the both left and right side of the inequality. |
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Solve given linear inequality.
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Systems
of linear inequalities
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The solution of the system of
simultaneous inequalities is the intersection of sets
of the individual solutions.
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92. |
Solve and graph the
solution of the given system of simultaneous inequalities.
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the solution of the system
is the open interval (3, + oo
). |
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Solving and graphing systems of
linear inequalities in two variables
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The set of points whose coordinates (x,
y)
satisfy the inequality ax+
by + c > 0 is a
half-plane of a Cartesian plane.
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93. |
Solve and graph the
solution of the given system of linear inequalities in two
variables.
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Solution:
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Plug the
coordinates of the origin |
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Solved
problems contents |
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