|
| 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, |
|
- 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 |
|
| The
solutions to the given equation are
x = - 8
and
x = 9. |
|
|
Example:
|
-3
- x
| = 5
|
|
Solution:
-3
-
x = 5
or
-3
-
x = -5 |
|
x = -
8
x = 2 |
|
| The
solutions to the given equation are
x = - 8
and
x = 2. |
|
|
Example:
|
x + 1 | =
2x -
3
|
|
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 |
|
| 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. |
|
|
Example:
|
x + 2 | =
| 2x -
5 |
|
|
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 |
|
| Check
solutions: |
| x =
7 =>
|
x + 2 | =
| 2x -
5 |, x = 1
=>
|
x + 2 | =
| 2x -
5 | |
|
|
7 + 2 | =
| 2 · 7 -
5
|
|
1 + 2 | = | 2 ·
1 -
5 |
|
|
9 = 9
3 = | -3 | |
|
| Therefore,
the
solutions to the given equation are x =
7 and x =
1. |
|
