Question

$x$ is a real number. Is $|x|<3$?
Statement A
A. $x(x+3)<0$
Statement B
B. $x(x-3)>0$

• A. The question can be answered by one of the statements alone but not by the other.
• B. The question can be answered by using either statement alone.
• C. The question can be answered by using both the statements together, but cannot be answered by using either statement alone.
• D. The question cannot be answered even by using both statements together.

Hint:

When testing inequalities, solve each separately and then check intersections when combining statements.

Answer 1

The Correct Option is C

Statement A:

Given: \[ x(x + 3) < 0 \] Solve the inequality:

• Critical points: \( x = 0, x = -3 \)
• This is a quadratic inequality where the parabola opens upwards.
• The expression is negative between the roots:

\[ -3 < x < 0 \] Now, can we conclude \( |x| < 3 \)? Yes, because:

• In the interval \( -3 < x < 0 \), the values of \( |x| \) are positive but less than 3.
• For example, if \( x = -2 \), then \( |x| = 2 < 3 \).

BUT: We are told to assess if this statement **alone** is enough to **guarantee** \( |x| < 3 \) in general. Yes — in fact, from this alone we can conclude \( |x| < 3 \), since \( x \) is between \(-3\) and \(0\). (So this contradicts the original claim in your comment — see clarification below.)

Statement B:

Given: \[ x(x - 3) > 0 \] Solve the inequality:

• Critical points: \( x = 0, x = 3 \)
• The expression is positive when:
  • \( x < 0 \), or
  • \( x > 3 \)

This is a union of two disjoint intervals. On its own, this does **not** help us determine whether \( |x| < 3 \), because:

• If \( x < 0 \), it could still be \( x < -3 \Rightarrow |x| > 3 \)
• If \( x > 3 \), then clearly \( |x| > 3 \)

So Statement B **alone** is **not sufficient**.

 

Combining Both Statements:

 

• From Statement A: \( -3 < x < 0 \)
• From Statement B: \( x < 0 \) or \( x > 3 \)
• The intersection of both statements is: \[ -3 < x < 0 \]
• This implies that: \[ |x| < 3 \]

Hence, both statements together are sufficient to conclude \( |x| < 3 \).

 

Final Answer:

\[ \boxed{\text{Both statements are needed to conclude that } |x| < 3} \]