An algebra problem by Anik Mandal

Algebra Level 4

Find the number of solutions to the equation ∣ ⌊ x ⌋ − 2 x ∣ = 4 | \lfloor x \rfloor - 2x |= 4 .

Notation : ⌊ ⋅ ⌋ \lfloor \cdot \rfloor denotes the floor function .


The answer is 4.

Anik Mandal by Anik Mandal , 21, India

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2 solutions

Sabhrant Sachan
Jul 14, 2016

∣ ⌊ x ⌋ − 2 x ∣ = 4 ⟹ − x − { x } = − 4 , 4 x + { x } = 4 , − 4 Let x be equal to I + f , Where I is the integral part of x and 0 < f < 1 I + 2 f = 4 , − 4 For f = 0 → I = 4 , − 4 For f = 1 2 → I = 3 , − 4 No other values of f will give us an integer in the LHS Possible values of x : 4 , 3.5 , − 4 , − 4.5 Our answer is 4 |\lfloor x \rfloor -2x|=4 \implies -x-\{x\} = -4,4 \\ x+\{x\} = 4,-4 \\ \text{Let }x \text{ be equal to } \color{royalblue}I+\color{#3D99F6}f , \text{Where } \color{royalblue}I \text{ is the integral part of x and } 0<\color{#3D99F6}f<1 \\ \color{royalblue}I+2\color{#3D99F6}f=4,-4 \\ \text{For } \color{#3D99F6}f=0 \rightarrow \color{royalblue}I=4,-4 \\ \text{For } \color{#3D99F6}f=\dfrac{1}2 \rightarrow \color{royalblue}I=3,-4 \\ \text{No other values of }f \text{ will give us an integer in the LHS} \\ \text{Possible values of } x : 4,3.5,-4,-4.5 \\ \text{Our answer is 4}

4 Helpful 2 Interesting 2 Brilliant 0 Confused

But we can't rule out the possibility of x tending to - 5 (or just less than -5) and also x tending to 3 (or just less than 3) Hence the actual answer should be 6

-5.00000....1, -4.5, - 4, 2.99999999......, 3.5, 4

Neil Shah - 2 years, 3 months ago

There's two combinations about the two absolute values. Two possible values of X, and two possible values of function. 2x2=4

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Not quite. The floor function can do weird things to the number of roots.

Calvin Lin Staff - 4 years, 11 months ago

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