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# Quadratic Absolute Value Inequality Example

- Question
- What is the solution set for x in the inequality |x
^{2}- 8| < 1?

- Solution
- if | x
^{2}- 8 | < 1 we have two possibilities: - -1 < x
^{2}- 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 < x
^{2}- 8 and x^{2}- 8 < 1 - case 1:
- x
^{2}- 8 > -1 (move the number -8 to the right side of the inequality) - x
^{2}> 8 - 1 - x
^{2}> 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:
- x
^{2}- 8 < 1 (move the number -8 to the right side of the inequality) - x
^{2}< 8 + 1 - x
^{2}< 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