Solve |x²-3|<3 . This looks simple but i could not get the right answer. The answer is (-√5,-1)U(1,√5). How to solve this inequality?

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Answer 1 · 2015-07-15T07:00:38

The solution is that the inequality should be $abs(x^2-3) < color(red)(2)$

As usual with absolute values, split into cases:

Case 1: $x^2 - 3 < 0$

If $x^2 - 3 < 0$ then $abs(x^2-3) = -(x^2-3) = -x^2+3$
and our (corrected) inequality becomes:

$-x^2+3 < 2$

Add $x^2-2$ to both sides to get $1 < x^2$

So $x in (-oo,-1) uu (1, oo)$

From the condition of the case we have

$x^2 < 3$ , so $x in (-sqrt(3), sqrt(3))$

Hence:

$x in (-sqrt(3), sqrt(3)) nn ((-oo,-1) uu (1, oo))$

$= (-sqrt(3), -1) uu (1, sqrt(3))$

Case 2: $x^2 - 3 >= 0$

If $x^2 - 3 >= 0$ then $abs(x^2-3) = x^2+3$ and our (corrected) inequality becomes:

$x^2-3 < 2$

Add $3$ to both sides to get:

$x^2 < 5$ , so $x in (-sqrt(5), sqrt(5))$

From the condition of the case we have

$x^2 >= 3$ , so $x in (-oo, -sqrt(3)] uu [sqrt(3), oo)$

Hence:

$x in ((-oo, -sqrt(3)] uu [sqrt(3), oo)) nn (-sqrt(5), sqrt(5))$

$= (-sqrt(5), -sqrt(3)] uu [sqrt(3), sqrt(5))$

Combined:

Putting case 1 and case 2 together we get:

$x in (-sqrt(5), -sqrt(3)] uu (-sqrt(3), -1) uu (1, sqrt(3)) uu [sqrt(3), sqrt(5))$

$=(-sqrt(5), -1) uu (1, sqrt(5))$