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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