Interval of x x

Algebra Level 3

x 2 + 2 x + 1 − x 2 − 4 x + 4 = 3 \large \sqrt{x^2+2x+1} - \sqrt{x^2-4x+4} = 3

What is the range of x x , where the above equation is true?

x ∈ { 1 , − 1 } x\in \{1,-1\} x ∈ [ 1 , ∞ ) x\in [1,\infty) x ∈ [ 1 , 2 ] x\in [1,2] x ∈ { 2 } x\in \{2\} x ∈ [ 2 , ∞ ) x\in [2,\infty) x ∈ ( 2 , ∞ ) x\in (2,\infty)

Vilakshan Gupta by Vilakshan Gupta , 19, India

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1 solution

Let f ( x ) = x 2 + 2 x + 1 − x 2 − 4 x + 4 f(x) = \sqrt{x^2+2x+1} - \sqrt{x^2-4x+4} . We note that f ( x ) f(x) is equivalent to the following:

f ( x ) = ∣ x + 1 ∣ − ∣ x − 2 ∣ = { − x − 1 + x − 2 = − 3 for x < − 1 x + 1 + x − 2 = 2 x − 1 for − 1 ≤ x < 2 x + 1 − x + 2 = 3 for x ≥ 2 f(x) = |x+1|-|x-2| = \begin{cases} -x-1 + x - 2 = - 3 & \text{for }x < - 1 \\ x+1 + x - 2 = 2x - 1 & \text{for }-1 \le x < 2 \\ x+1-x+2 = 3 & \text{for }x \ge 2 \end{cases}

Therefore, the equation is true for x ∈ [ 2 , ∞ ) \boxed{x \in [2, \infty)} .

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