how to solve this question? √(x+1)+√(x-1)÷√(x+1)-√(x-1)=3÷2

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Solve `(sqrt(x+1)+sqrt(x-1))/(sqrt(x+1)-sqrt(x-1))=3/2`

The product of the means equals the product of the extemes (so you can "cross multiply"):

`3sqrt(x+1)-3sqrt(x-1)=2sqrt(x+1)+2sqrt(x-1)` Collect like terms

`sqrt(x+1)=5sqrt(x-1)` Square both sides

`x+1=25(x-1)`

`x+1=25x-25`

`24x=26`

`x=13/12`

Since we squared both sides you must check the answer. (Squaring can introduce extraneous solutions.)

`(sqrt(25/12)+sqrt(1/12))/(sqrt(25/12)-sqrt(1/12))` Multiply numerator and denominator by `1/sqrt(12)`

`(sqrt(25)+sqrt(1))/(sqrt(25)-sqrt(1))=6/4=3/2` as required.

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to multiplicate both numerator and denominator by the conjugate expression `sqrt(x+1) + sqrt(x-1)` to remove the square roots from denominator, such that:

`((sqrt(x+1) + sqrt(x-1))(sqrt(x+1) + sqrt(x-1)))/((sqrt(x+1) - sqrt(x-1))(sqrt(x+1) + sqrt(x-1))) = 3/2`

You need to convert the product `((sqrt(x+1) - sqrt(x-1))(sqrt(x+1) + sqrt(x-1)))` into a difference of squares, such that:

`(sqrt(x+1) + sqrt(x-1))^2/(x + 1 - x + 1) = 3/2`

Reducing like terms, yields:

`(sqrt(x+1) + sqrt(x-1))^2/2 = 3/2`

Reducing duplicate factors yields:

`x + 1 + 2sqrt(x^2-1) + x - 1 = 3`

`2x + 2sqrt(x^2-1) = 3 => x + sqrt(x^2-1) = 3/2`

You need to isolate the square root to the left side, such that:

`sqrt(x^2-1) = 3/2 - x`

Raising to square both sides, yields:

`x^2-1 = 9/4 - 3x + x^2`

Reducing duplicate terms both sides, yields:

`-1 = 9/4 - 3x => 3x = 1 + 9/4`

`3x = 13/4 => x = 13/12`

Hence, evaluating the solution to the given equation yields `x = 13/12.`

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