Absolute value inequalities
      Solving linear inequalities with absolute value
Absolute value inequalities
Solving linear inequalities with absolute value
Graphical interpretation of the definition of the absolute value of a function = 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  = - f (x) is obtained by reflecting the graph of = f (x) across the x-axis.
Hence, the graph of the absolute value of the function = x, i.e., | x | is
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  = 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  = 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 [-1, 5]

Generally, when solving inequalities involving the absolute value of a linear function = 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 <   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.
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.
The same result can be obtained algebraically by solving the two inequalities
Intermediate algebra contents
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