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Absolute Valued Functions

When we look at 2 and -2, we notice that they have the same numerical value of 2. The absolute value of a number x, denoted by |x|, is the numerical value of x. 

Thus we have |2| = 2 and |-2|=2.

Example 1

Solve the equation |x + 1| = 2x - 3

|x + 1| = 2x - 3

  x + 1 = 2x - 3        or         x + 1 = 3 - 2x
        x = 4              or               x = 2/3 (n.a.)
x = 4

Example 2

Solve the equation |2x + 4| =x2 + 1

 
2x + 4 = x2 + 1   or            2x + 4 = -x2 - 1
x2 - 2x - 3 = 0         or            x2 + 2x + 5 = 0 (n.a.)
(x + 1)(x - 3)
x = -1 or 3

Example 3

Given that f(x) = |3x - 2|, find

(a) the value of f(-1) and f(2),
(b) the values of x for which f(x) = 8,
(c) the value of x for which f(x) = 8.
 
(a) f(-1) = |3 x (-1) -2| = |-5| = 5
      f(2) = |3 x 2 -2| = |4| = 4
 
(b) f(x) = 8
|3x - 2| = 8
3x - 2 = -8        or        3x - 2 = 8
3x = -6             or             3x = 10
x = -2 or 10/3
 
(c) f(x) = x
|3x - 2| = x
3x - 2 = x         or        3x - 2 = -x
4x = 2              or             2x = 2
x = ½ or 1.