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Quadratic Absolute Value Inequality Example

Question
What is the solution set for x in the inequality |x2 - 8| < 1?
Solution
if | x2 - 8 | < 1 we have two possibilities:
-1 < x2 - 8 < 1
formula used: if a > 0, then |f(x)| < a is equivalent to -a < f(x) < a
we get a group of inequality: -1 < x2 - 8 and x2 - 8 < 1
case 1:
x2 - 8 > -1 (move the number -8 to the right side of the inequality)
x2 > 8 - 1
x2 > 7 (square both side of the inequality)
there are two solutions: x < - square root of 7 or x > square root of 7
square root of 7 = 2.65
the solution in case1 is: x < -2.65 or x > 2.65
case 2:
x2 - 8 < 1 (move the number -8 to the right side of the inequality)
x2 < 8 + 1
x2 < 9
then -3 < x < 3
the solution in case2 is: -3 < x < 3
therefore, the solution for the given inequality is: -3 < x < -2.65 and 2.65 < x < 3.
absolute inequality formula:
if a > 0, |f(x)| < a is equivalent to -a < f(x) < a
if a > 0, |f(x)| > a is equivalent to f(x) < -a or f(x) > a