Log1/2(x²-4)>log1/2(x+2)-1

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To solve this inequality, we need to first simplify both sides by using the properties of logarithms.

Simplifying the left side:
log1/2(x²-4) = log1/2((x-2)(x+2))
= log1/2(x-2) + log1/2(x+2)

Simplifying the right side:
log1/2(x+2) - 1 = log1/2(x+2) - log1/2(2)
= log1/2(x+2/2)
= log1/2(x)

Now, our inequality becomes:
log1/2(x-2) + log1/2(x+2) > log1/2(x)

Combine the logarithms on the left side:
log1/2((x-2)(x+2)) > log1/2(x)

Since the bases are the same, we can drop the logarithms:
(x-2)(x+2) > x

Expanding the left side:
x² - 4 > x

This is a quadratic inequality, so we need to set it equal to zero to find the critical points:
x² - 4 - x > 0
x² - x - 4 > 0

Now we can factor the quadratic equation:
(x-2)(x+2) > 0

The critical points are x=2 and x=-2. We can now test the intervals (-inf,-2), (-2,2), and (2,inf) to determine where the inequality holds true.

Testing the interval (-inf, -2):
Pick x=-3:
(-3-2)(-3+2) = (-5)(-1) = 5 > 0, which is true.

Testing the interval (-2,2):
Pick x=0:
(0-2)(0+2) = (-2)(2) = -4 > 0, which is false.

Testing the interval (2, inf):
Pick x=3:
(3-2)(3+2) = (1)(5) = 5 > 0, which is true.

Therefore, the solution to the inequality is x in (-∞, -2) U (2, ∞).