back to Algebra
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