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Question
Using properties of proportion, solve for x:
`(sqrt(x + 1) + sqrt(x - 1))/(sqrt(x + 1) - sqrt(x - 1)) = (4x - 1)/2`
Solution
`(sqrt(x + 1) + sqrt(x - 1))/(sqrt(x + 1) - sqrt(x - 1)) = (4x - 1)/2`
Applying componendo and dividendo,
`(sqrt(x + 1) + sqrt(x - 1) + sqrt(x + 1) - sqrt(x - 1))/(sqrt(x + 1) + sqrt(x - 1) - sqrt(x + 1) + sqrt(x - 1)) = (4x - 1 + 2)/(4x - 1 - 2)`
`(2sqrt(x + 1))/(2sqrt(x - 1)) = (4x + 1)/(4x - 3 )`
Squaring both sides,
`(x + 1)/(x - 1) = (16x^2 + 1 + 8x )/(16x^2 + 9 - 24x)`
Applying componendo and dividendo,
`(x + 1 + x - 1)/(x + 1 - x + 1) = (16x^2 + 1 + 8x + 16x^2 + 9 - 24x)/(16x^2 + 1 + 8x - 16x^2 - 9 + 24x)`
`(2x)/2 = (32x^2 + 10 - 16x )/(32x - 8)`
`x = (16x^2 + 5 - 8x)/(16x - 4)`
16x2 – 4x = 16x2 + 5 – 8x
4x = 5
`x = 5/4`
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