Question

Which of the following inequalities holds true?
(A) \(\sqrt{5} + \sqrt{3}>\sqrt{6} + \sqrt{2}\)
(B) If \(a>b\) and \(c<0\), then \(\frac{a}{c}<\frac{b}{c}\)
(C) \(\frac{1}{x^2}>\frac{1}{x}>1\), if \(0<x<1\)
(D) If a and b are positive integers and \(\frac{a-b}{6.25} = \frac{4}{2.5}\) then \(b>a\)
Choose the correct answer from the options given below:

• A. (A), (B) and (D) only
• B. (A), (B) and (C) only
• C. (A) and (B) only
• D. (B) and (C) only

Hint:

When testing inequalities with variables, plugging in a simple test value that fits the condition (like \(x=0.5\) for \(0<x<1\)) is a quick way to check its validity. For inequalities with square roots, squaring both sides is often the most effective method, but remember this only works if both sides are non-negative.

Answer 1

The Correct Option is B

Objective: Evaluate the truthfulness of four mathematical statements.

Statement Analysis:

Statement (A): \( \sqrt{5} + \sqrt{3} > \sqrt{6} + \sqrt{2} \)

Squaring both sides yields \( 8 + 2\sqrt{15} \) on the left and \( 8 + 2\sqrt{12} \) on the right. Since \( \sqrt{15} > \sqrt{12} \), this statement is true.

Statement (B): For \( a > b \) and \( c < 0 \), it is stated that \( \frac{a}{c} < \frac{b}{c} \). This is consistent with the rule that dividing by a negative number reverses inequality direction. Thus, the statement is true.

Statement (C): For \( 0 < x < 1 \), the inequality \( \frac{1}{x^2} > \frac{1}{x} > 1 \) is examined. Testing with \( x = 0.5 \) gives \( 4 > 2 > 1 \), confirming the statement's validity. It is true.

Statement (D): Given positive integers \( a \) and \( b \), and the equation \( \frac{a-b}{6.25} = \frac{4}{2.5} \), it is asserted that \( b > a \). Solving the equation results in \( a - b = 10 \), which implies \( a > b \). Therefore, this statement is false.

Conclusion: Statements (A), (B), and (C) are true. Statement (D) is false. The correct selection includes only (A), (B), and (C).