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Absolute value
equations |
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Solving absolute value
equations |
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Linear Inequalities
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Solving
inequalities |
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Properties of
inequalities |
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Examples
of solving single linear inequalities |
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Solving
compound (double) inequalities |
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Absolute value
inequalities |
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Solving
linear inequalities with absolute value |
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Examples
of solving
linear inequalities with absolute value |
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| Absolute value
equations |
| Solving absolute value
equations |
| Recall
that the absolute value of a real number a,
denoted |
a |,
is the number without its
sign and represents the distance between 0 (the origin) and that
number on the real number line. |
|
Thus, regardless of the value of a number
a
its absolute value is
always either positive or zero, never negative that is, | a
|
> 0.
|
| To
solve an absolute value equation, isolate the absolute value on
one side of the equation, and use the definition of absolute
value. |
| If the number on the other side of the equal sign is positive, we will need to set up two
equations to get rid of the absolute value, |
|
- the first equation that set the expression inside the absolute value symbol equal to the other side of the equation, |
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- and the second equation that set the expression inside the absolute value equal to the
opposite of the number on the other side of the equation. |
| Solve
the two equations and verify solutions by plugging the solutions
into the original equation. |
| If
the number on the other side of the absolute value equation is negative then
the equation has no solution. |
|
|
Example:
|
1
- 2x
| = 17
|
|
Solution:
1
-
2x =
17
or
1 -
2
x = -17 |
|
2
x = - 16
2x =
18 |
|
x = -
8
x = 9 |
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| The
solutions to the given equation are
x = - 8
and
x = 9. |
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Example:
|
-3
- x
| = 5
|
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Solution:
-3
-
x = 5
or
-3
-
x = -5 |
|
x = -
8
x = 2 |
|
| The
solutions to the given equation are
x = - 8
and
x = 2. |
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Example:
|
x + 1 | =
2x -
3
|
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Solution:
x + 1 = 2x
-
3
or
x + 1 = -(2
x -
3) |
|
x - 2x
= - 4
x +
2x = 3 -
1 |
|
x = 4
3x = 2
=> x = 2/3 |
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| Check
solutions: |
| x = 4
=>
|
x + 1 | =
2x -
3,
x = 2/3
=>
|
x + 1 | =
2x -
3 |
|
|
4
+ 1 | =
2 · 4 -
3
|
2/3 + 1 | =
2 · 2/3 -
3
|
|
5 = 5
5/3 is not equal
-5/3 |
| The
check shows that x =
2/3 is not a solution,
because the right side of the equation becomes negative. There is a single solution to this equation: x = 4. |
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|
Example:
|
x + 2 | =
| 2x -
5 |
|
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Solution: As both
sides of the equation contain absolute values the
only way the two sides are equal is, the two quantities
inside the absolute value bars are equal or equal but with
opposite signs.
|
|
x + 2 = 2x
-
5
or
x + 2 = -(2
x -
5) |
|
x -
2x
= -
5 -
2
x + 2 = -2
x +
5 |
|
-x =
-7
3x = 3 |
|
x = 7
x = 1 |
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| Check
solutions: |
| x =
7 =>
|
x + 2 | =
| 2x -
5 |, x = 1
=>
|
x + 2 | =
| 2x -
5 | |
|
|
7 + 2 | =
| 2 · 7 -
5
|
|
1 + 2 | = | 2 ·
1 -
5 |
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|
9 = 9
3 = | -3 | |
|
| Therefore,
the
solutions to the given equation are x =
7 and x =
1. |
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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. |
| 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: |
| -
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: |
|
| 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. |
|
| 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 |
|
 |
|
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 |
|
- |
0 |
+ |
+ |
+ |
|
| x
- 3 |
|
- |
- |
- |
0 |
+ |
|
| (x
- 3)(x
+ 2) |
|
+ |
0 |
- |
0 |
+ |
|
|
|
| Thus,
the given inequality is satisfied for -
oo
< x
<
- 2
or 3
<
x
< +
oo |
|
in the interval notation (
- oo
, -2
]
U [
3, +
oo
) |
 |
|
| Solving
compound (double) inequalities |
| Use the same procedure to solve a compound inequality as for solving single inequalities. |
| Example:
-4
< 2(x
- 3)
< 5 |
|
|
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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| 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. |
| The
definition for the absolute value of a function is given by |
|
|
| Thus,
for
values of x
for which f(x)
is nonnegative, the graph of | f(x)| is the
same as that of f(x).
|
| For values of
x
for which f(x)
is
negative, the graph of | f(x)
| is a reflection of
the graph of f(x)
on the x
axis. |
| That
is, the
graph of y = -
f(x) is obtained by reflecting the graph
of y = f(x)
across the x-axis. |
| Hence,
the graph of the absolute value of
the function
y =
x,
i.e.,
| x | is |
 |
|
| Examples
of solving
linear inequalities with absolute value |
| Example:
Solve the absolute value
inequality | x
-
2
| <
3. |
|
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
| is the
same as that of y =
x -
2.
|
| For values of
x
for which y
is
negative, the graph of | y
| is a reflection of
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. |
 |
| The
graph shows that
values of x from
the closed interval [-1,
5] satisfy the given inequality. |
| The
same result can be obtained algebraically by solving the
compound inequality. |
 |
 |
| x
Î [-1,
5] |
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Generally, when
solving inequalities involving the absolute value of a linear
function y =
ax+
b, we consider the two different
cases.
|
| 1)
When the initial inequality is of the form
| ax +
b |
< c
or ( | ax +
b |
< c ) |
| we
replace it by the compound inequality -
c
<
ax +
b <
c
or (-
c
< ax +
b < c). |
| Note that there are two inequalities
here and the way they are written implies that both first AND
the second inequalities must be satisfied. The solution set will
then be an intersection of two
sets, as is shown above.
|
| 2)
When the initial inequality is of the form
| ax
+
b |
> c
or ( | ax
+
b | > c) |
| we
replace it by two inequalities ax
+
b < -
c OR ax
+
b > c or
(
ax +
b
< - c
OR ax
+
b
> c) |
| The way
the two inequalities are written implies that the first OR the
second inequality must be satisfied. |
| The solution set will then
be the union of two sets, as is
shown in the example below. |
| Notice that in both cases the corresponding graph of the given inequality suggests its solution. |
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|
| Example:
Solve the absolute value
inequality | x
-
2
| > 3. |
|
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. |
| The
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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