Solve the inequation 8> x^2 – 4 > 4?

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We have to solve the inequation 8> x^2 – 4 > 4.

Now, let us divide the given inequation into two separate inequations.

- 8 > x^2 – 4

=> 8 + 4 > x^2

=> 12 > x^2

=> sqrt 12 > x > -sqrt 12

- x^2 – 4 > 4

=> x^2 > 4 + 4

=> x^2 > 8

=> -sqrt 8 > x > sqrt 8

Now, to meet the conditions we have identified above, x should be less than sqrt 12 and greater than sqrt 8 or greater than -sqrt 12 and less than - sqrt 8.

**Therefore x can only take values that lie in ( - sqrt 12, -sqrt 8) and ( sqrt 8 , sqrt 12).**

4 < x^2 - 4 < 8

Add 4 to both sides.

==> 8 < x^2 < 12

Now let us separate and solve each side.

==> 8 < x^2 and x^2 < 12

Let us solve 8 < x^2

Take the square root of both sides.

==> sqrt8 < x and sqrt8 < -x

**==> 2sqrt2 < x and -2sqrt2 > x**

**==> x = ( 2sqrt2,inf) U ( -inf, -2sqrt2)**

Now the other side.

==> x^2 < 12

==> x < sqrt12 OR -x < sqrt12

==> x < 2sqrt3 OR x > -2sqrt3

**==> x = ( 2sqrt3, inf) U ( -inf, -2sqrt3) **

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