Ooh! This looks complicated, but is it really complicated?

Algebra Level 3

$\dfrac{6x^2 - 2(x+1) \sqrt{x^2-1} + 3x-6}{x+1 -\sqrt{x-1} - \sqrt{2-x} - \sqrt{2(x^2+2)}} \leq 0$

Find the interval of $x$ that satisfy the inequality above.

1 \leq x \leq \frac {5}{4}

1 \leq x \leq 2

x \leq \frac {5}{4}

x \geq 0

1 solution

Dan Brabec
Apr 15, 2021

The values for x when the statement is less than 0 (negative) are bound by the square roots of 2-x and x-1 from the denominator. When x is less than 1, the value of the expression is no longer a real number. This is also true when x > 2. This limits the possible outcomes to values between 1 and 2 inclusive.

Within that range of values, the numerator will always be positive, and the denominator will always be negative so the resulting expression will always be negative.

Ooh! This looks complicated, but is it really complicated?

1 solution

0 pending reports