Answer:
False
Explanation:
Given expression: x/(x - 3) + (x - 3)/x
First, we need to find a common denominator to combine the two fractions. The common denominator is x(x - 3):
x/(x - 3) = (x * x)/[x * (x - 3)] = x^2 / (x^2 - 3x)
(x - 3)/x = (x - 3) * (x - 3)/[x * (x - 3)] = (x^2 - 6x + 9) / (x^2 - 3x)
Now, the expression becomes:
x^2 / (x^2 - 3x) + (x^2 - 6x + 9) / (x^2 - 3x)
Combine the fractions with the common denominator:
(x^2 + x^2 - 6x + 9) / (x^2 - 3x)
Simplify the numerator:
(2x^2 - 6x + 9) / (x^2 - 3x)
Now we are given that this expression is equal to 5/2:
(2x^2 - 6x + 9) / (x^2 - 3x) = 5/2
Cross-multiply:
2 * (2x^2 - 6x + 9) = 5 * (x^2 - 3x)
Simplify both sides:
4x^2 - 12x + 18 = 5x^2 - 15x
Subtract 4x^2 and add 15x to both sides:
18 = x^2
Take the square root of both sides:
x = ±√18
Simplify the square root:
x = ±3√2
So, the solutions for x are x = 3√2 and x = -3√2.
However, when you plug these solutions back into the original expression, you'll find that they do not satisfy the equation x/(x - 3) + (x - 3)/x = 5/2. Therefore, there seems to be an error or misunderstanding in the equation provided. The equation does not hold true for the given expression.