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Question

Mathematics Question on Inequalities

If x is a positive number, then which of the following fractions has the greatest value?

A

xx\frac{x}{x}

B

x+1x\frac{x+1}{x}

C

x(x+1)\frac{x}{(x+1)}

D

(x+2)(x+3)\frac{(x+2)}{(x+3)}

Answer

x+1x\frac{x+1}{x}

Explanation

Solution

The correct option is (B): x+1x\frac{x+1}{x}
Explanation: The given options are:
- xx=1\frac{x}{x} = 1

- x+1x=1+1x\frac{x+1}{x} = 1 + \frac{1}{x}

- xx+1\frac{x}{x+1}

- x+2x+3\frac{x+2}{x+3}

Since xx is a positive number, let's compare the values of these fractions:
- xx=1\frac{x}{x} = 1

- x+1x=1+1x\frac{x+1}{x} = 1 + \frac{1}{x}, which is always greater than 1.

- xx+1\frac{x}{x+1}, which is always less than 1 for any positive xx.

- x+2x+3\frac{x+2}{x+3}, which is also less than 1 for any positive xx.

Clearly, x+1x\frac{x+1}{x} has the greatest value because it is always greater than 1, while the other fractions are less than or equal to 1.

Thus, the correct answer is B: x+1x\frac{x+1}{x}.