Equality is fun

Algebra Level 3

If the equation ∣ 2 − x ∣ − ∣ x + 1 ∣ = k |2-x|-|x+1|=k has exactly one solution, then find the sum of all real integer values of k k .

Notation: ∣ ⋅ ∣ | \cdot | denotes the absolute value function .


The answer is 0.

Amir Raza by Amir Raza , 21, India

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1 solution

Sabhrant Sachan
Dec 24, 2016

Let f ( x ) = ∣ 2 − x ∣ − ∣ x + 1 ∣ f(x) = |2-x|-|x+1|

f ( x ) = ( if x < − 1 f ( x ) = 2 − x + x + 1 = 3 if − 1 ≤ x < 2 f ( x ) = 2 − x − x − 1 = 1 − 2 x if x ≥ 2 f ( x ) = − 2 + x − x − 1 = − 3 ) f(x) = \left( \begin{matrix} \text{if } x<-1 & f(x) =2-\cancel{x}+\cancel{x}+1 = 3 \\ \text{if } -1 \le x < 2 & f(x) =2-x-x-1 = 1-2x \\ \text{if } x \ge 2 & f(x) =-2+\cancel{x}-\cancel{x}-1 = -3 \end{matrix} \right)

Unique value of k k is present only when − 1 < x < 2 -1 <x < 2

1 − 2 x = k − 1 < x = 1 − k 2 < 2 − 3 < k < 3 1-2x = k \\ -1< x=\dfrac{1-k}{2} < 2 \\ -3<k<3

possible values of k = 2 , 1 , 0 , − 1 , − 2 k = 2,1,0,-1,-2

Sum of all possible values: 0 \boxed{0}

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