Is that too big?

Algebra Level 2

Solve for x x in the given expression:

1 1 1 x + 1 2 + 1 1 x + 1 2 + 1 1 1 x − 1 2 + 1 1 x − 1 2 = 4 3 \Large\frac{1}{\frac{1}{\frac{1}{x}+\frac{1}{2}}+\frac{1}{ \frac{1}{x}+\frac{1}{2}}}+\frac{1}{\frac{1}{\frac{1}{x}-\frac{1}{2}}+\frac{1}{\frac{1}{x}-\frac{1}{2}}}=\frac{4}{3}

If the solution is A B \frac{A}{B} , where A A and B B are coprime positive integers, then what is A + B A+B ?


The answer is 7.

Hana Wehbi by Hana Wehbi , 51, USA

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2 solutions

Chew-Seong Cheong
Jun 19, 2018

1 1 1 x + 1 2 + 1 1 x + 1 2 + 1 1 1 x − 1 2 + 1 1 x − 1 2 = 4 3 1 2 1 x + 1 2 + 1 2 1 x − 1 2 = 4 3 1 2 2 + x 2 x + 1 2 2 − x 2 x = 4 3 1 4 x 2 + x + 1 4 x 2 − x = 4 3 2 + x + 2 − x 4 x = 4 3 4 4 x = 4 3 1 x = 4 3 ⟹ x = 3 4 \begin{aligned} \frac 1{\frac 1{\frac 1x+\frac 12} + \frac 1{\frac 1x+\frac 12}} + \frac 1{\frac 1{\frac 1x-\frac 12} + \frac 1{\frac 1x-\frac 12}} & = \frac 43 \\ \frac 1{\frac 2{\frac 1x+\frac 12}} + \frac 1{\frac 2{\frac 1x-\frac 12}} & = \frac 43 \\ \frac 1{\frac 2{\frac {2+x}{2x}}} + \frac 1{\frac 2{\frac {2-x}{2x}}} & = \frac 43 \\ \frac 1{\frac {4x}{2+x}} + \frac 1{\frac {4x}{2-x}} & = \frac 43 \\ \frac {2+x+2-x}{4x} & = \frac 43 \\ \frac 4{4x} & = \frac 43 \\ \frac 1x & = \frac 43 \\ \implies x & = \frac 34 \end{aligned}

Therefore, A + B = 3 + 4 = 7 A+B = 3+4 = \boxed{7} .

3 Helpful 0 Interesting 0 Brilliant 0 Confused
X X
Jun 18, 2018

By simplfying,we get 1 x = 4 3 , x = 3 4 \dfrac1x=\dfrac43,x=\dfrac34

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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