Question

Is \( \frac{5x^2 + 2}{25}<1 \)?
(1) \( 5x^2<1 \)
(2) \( x<0 \)

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are not sufficient

Hint:

When solving inequalities, isolate the variable on one side and check if the given statements provide enough information to conclude the solution.

Answer 1

The Correct Option is A

Step 1: Analyze statement (1).
From statement (1), we have \( 5x^2<1 \). Dividing both sides by 5: \[ x^2<\frac{1}{5} \] Now, substitute this into the original inequality: \[ \frac{5x^2 + 2}{25}<1 \quad \implies \quad 5x^2 + 2<25 \quad \implies \quad 5x^2<23 \] Since statement (1) already provides us with the condition \( 5x^2<1 \), which is less than 23, we can conclude that the inequality \( \frac{5x^2 + 2}{25}<1 \) holds. Therefore, statement (1) alone is sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that \( x<0 \), but we cannot conclude whether the inequality holds for all negative values of \( x \) without knowing the specific value of \( x \). Thus, statement (2) alone is not sufficient.
\[ \boxed{A} \]