Solve for X. 2x-1/3x+1 +X+1/x-1 = 5/2 where X is not equals to 1 and -1/3
Answers
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Given,
(2x-1/3x+1) + (x+1/x-1) = 5/2 where x ≠ 1 and -1/3
To find,
The value of x.
Solution,
The value of x will be 3 and -3/5.
We can easily solve this problem by following the given steps.
According to the question,
(2x-1/3x+1) + (x+1/x-1) = 5/2
Taking the L.C.M. of the denominators,
(2x-1) (x-1) + (x+1) (3x+1)/ (3x+1) (x-1) = 5/2
( Note that when we add the two fractions, the L.C.M. is divided by the denominator and the quotient is multiplied by the numerator.)
Now, further solving the brackets by multiplying,
2x² - 2x - x + 1 + 3x² + x + 3x + 1/(3x² - 3x + x -1) = 5/2
2x² - 3x + 1 + 3x² + 4x + 1/(3x² - 2x -1) = 5/2 [ Adding the numbers with variable x]
( 5x² + x + 2)/ (3x² - 2x - 1) = 5/2
Using the cross multiplication method,
2(5x² + x + 2) = 5(3x² - 2x - 1)
10x² + 2x + 4 = 15x² - 10x - 5
10x² - 15x² + 2x + 10x + 4 + 5 = 0 ( Moving all the numbers from the right-hand side to the left-hand hand side will result in the change of their sign from minus to plus or plus to minus.)
-5x² + 12x + 9 = 0
Taking minus common from all the numbers,
5x² - 12x - 9 = 0
Now, factorising it by splitting the middle term such that their multiplication will be (5x² × 9) and subtraction will be -12x.
5x² - 15x + 3x - 9 = 0
5x(x-3) +3(x-3) = 0 [Taking 5x common from the first two terms and 3 commons from the last two terms.]
(x-3) (5x+3) = 0 [Taking (x-3) common from the expression.]
(x-3) = 0 [Eqauting both the brackets with 0.]
x = 3
(5x+3) = 0
5x = -3
x = -3/5
Hence, the values of x are 3 and -3/5.