Question

Is \(x\) positive? (1) \(x^2-9=0\) (2) \(x^3-27=0\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. \(\quad 1\) and 2 together are not sufficient

Short answer

B. 2 alone, not 1 alone

Step-by-step solution

Analyze Statement (1)

Solve the equation from statement (1): \( x^2 - 9 = 0 \). This can be factored as: \( (x - 3)(x + 3) = 0 \). The solutions are: \( x = 3 \) or \( x = -3 \). Since one solution is positive and the other is negative, statement (1) alone is not sufficient to determine if \( x \) must be positive.

Analyze Statement (2)

Solve the equation from statement (2): \( x^3 - 27 = 0 \). This can be rewritten as: \( x^3 = 27 \). Taking the cube root of both sides gives us: \( x = 3 \). Since the only solution is positive, statement (2) alone is sufficient to determine that \( x \) is positive.

Combine Both Statements

Although it was not required, combining both statements confirms the findings: Statement (1) gives \( x = 3 \) or \( x = -3 \), and statement (2) gives \( x = 3 \). Since we already concluded the sufficiency from statement (2), combining both would still result in the same conclusion that \( x \) is positive.

Key concepts

algebraic equations

Algebraic equations are mathematical statements indicating that two expressions are equal. In the given exercise, we are provided with two algebraic equations to determine if the variable \( x \) is positive.

First, let’s look at the equation from the first statement: \( x^2 - 9 = 0 \). We can solve this by factoring the quadratic equation as \( (x - 3)(x + 3) = 0 \). This gives us two potential solutions for \( x \):\( x = 3 \) or \( x = -3 \).

Notice that one solution is positive (\