Question - Simplification of a Complex Fraction
Solution:
To simplify the given expression, we need to combine the terms in the numerator and the denominator first and then simplify the complex fraction.$$\frac{\frac{2}{x}-5}{6 + \frac{3}{x}}$$First, let's find a common denominator for the terms in the numerator and the terms in the denominator.For the numerator, the common denominator is $$ x $$. For the denominator, it is also $$ x $$.Numerator:$$\frac{2}{x} - 5 = \frac{2}{x} - \frac{5x}{x} = \frac{2 - 5x}{x}$$Denominator:$$6 + \frac{3}{x} = \frac{6x}{x} + \frac{3}{x} = \frac{6x + 3}{x}$$Now that we have a common denominator, we can combine the two parts:$$\frac{\frac{2 - 5x}{x}}{\frac{6x + 3}{x}}$$Next, we can simplify by multiplying by the reciprocal of the denominator:$$\frac{2 - 5x}{x} \times \frac{x}{6x + 3}$$Notice that $$ x $$ in the numerator and denominator will cancel out:$$\frac{2 - 5x}{6x + 3}$$Now the expression is simplified:$$\frac{2 - 5x}{6x + 3}$$This is the simplified form of the given expression.
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