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Linear Inequalities
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Solving inequalities |
Properties of
inequalities |
Examples
of solving single linear inequalities |
Solving
compound (double) inequalities |
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Linear Inequalities
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A linear inequality
is one that can be reduced to the standard form
ax +
b
> 0
where a,
b
Î
R,
and where other inequality signs like
<, > and
< can appear. |
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Solving
inequalities |
The solutions to an inequality are all values of
x
that make the inequality true. Usually the answer is a range |
of
values of x
that we plot on a number line. |
We
use similar method to solve linear inequalities as
for solving linear equations: |
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simplify both sides, |
-
bring all the terms with the variable on one side and the
constants
on the other side, |
- and then multiply/divide both sides by the
coefficient of the variable to get the solution while applying |
following properties: |
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Properties of
inequalities |
1. Adding
or subtracting the same quantity from both sides of an
inequality will not change the direction of the inequality sign. |
2. Multiplying
or dividing both sides of an inequality by a positive number leaves the
inequality symbol unchanged. |
3.
Multiplying or dividing both sides of an inequality
by the same negative number, the sense of the inequality changes,
i.e., it reverses the direction of the inequality sign. |
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Examples
of solving single linear inequalities: |
Solve
each of the following inequalities, sketch the solution on the
real number line and express the solution |
in interval notation. |
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Example:
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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Example:
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Solution: -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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Example:
(x
- 3)
· (x
+ 2)
> 0 |
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 |
Use the same procedure to solve a compound inequality as for solving single inequalities. |
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Example:
-
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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Example: |
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Solution: |
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Intermediate
algebra contents |
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