The number of distinct real values of x, satisfying the equation max{x, 2} - min{x, 2} = |x + 2| - |x - 2|, is

Question

Let $ n(A) = m $ and $ n(B) = n $, if the number of subsets of $ A $ is 56 more than that of subsets of $ B $, then $ m + n $ is equal to:

Answer 1

Simplify the left side: For every real $x$, $\max\{x,2\} - \min\{x,2\} = |x-2|$. The equation becomes $|x-2| = |x+2| - |x-2|$, that is $2|x-2| = |x+2|$.

Remove the modulus by squaring: $4(x-2)^2 = (x+2)^2$.
\[4(x^2 - 4x + 4) = x^2 + 4x + 4\]\[3x^2 - 20x + 12 = 0\]

Solve the quadratic: The discriminant is $400 - 144 = 256$, so $x = \dfrac{20 \pm 16}{6}$, giving $x = 6$ or $x = \dfrac{2}{3}$.

Both values satisfy the original equation, so there are $2$ distinct real solutions.

Answer: $2$.

The number of distinct real values of x, satisfying the equation max{x, 2} - min{x, 2} = |x + 2| - |x - 2|, is

Correct Answer: 3

Solution and Explanation

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