Question

Is \(x\) is positive or negative? (1) \(x^2-5 x=-6\) (2) \(4 / x=x\) 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. 1 and 2 together are not sufficient

Short answer

A. 1 alone, not 2 alone

Step-by-step solution

- Solve Equation 1

First, we solve the equation given in statement (1): \[ x^2 - 5x = -6 \] Rearrange it to the standard quadratic form: \[ x^2 - 5x + 6 = 0 \]This can be factored into: \[ (x-2)(x-3) = 0 \]So, the solutions are: \[ x = 2 \] and \[ x = 3 \]Both solutions are positive.

- Interpret Equation 1

From solving statement (1), both solutions are positive. Therefore, statement (1) alone is sufficient to determine that \(\ x>0 \).

- Solve Equation 2

Now, we solve the equation given in statement (2): \[ \frac{4}{x} = x \]Multiply both sides by \( x \) (where \( x ≠ 0 \)): \[ 4 = x^2 \] Taking the square root of both sides: \[ x = 2 \] or \[ x = -2 \]

- Interpret Equation 2

From solving statement (2), the solutions are \( x = 2 \) or \( x = -2 \), which means \( x \) could be positive or negative. Therefore, statement (2) alone is not sufficient to determine the sign of \( x \).

- Combine Both Statements

Statement (1) alone is sufficient to determine that \( x \) is positive, and while statement (2) alone is not sufficient, we have already determined the sign of \( x \) using statement (1). Considering both together does not change the sufficiency determined by statement (1).

Key concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra. They are equations of the form \( ax^2 + bx + c = 0 \), where `a`, `b`, and `c` are constants, and `x` represents an unknown variable. In our example, given by the equation \( x^2 - 5x + 6 = 0 \), we can solve it by factoring it to \( (x-2)(x-3) = 0 \). This means the solutions to the equation are the values of `x` that make the product zero, namely \( x = 2 \) and \( x = 3 \).
These solutions can be found using various methods such as:

• Factoring: As demonstrated, rewriting the equation as a product of binomials.
• Quadratic formula: Using the formula \( x = \frac{{-b \, \text{±} \, \text{√}(b^2-4ac)}}{2a} \).
• Completing the square: Manipulating the equation to make it a perfect square trinomial.

Understanding how to solve quadratic equations is essential for tackling more complex algebra problems in GMAT and other standardized tests.

GMAT Quantitative Reasoning

The GMAT Quantitative Reasoning section tests your ability to analyze data and draw conclusions using reasoning skills. This section includes problems that require understanding of concepts such as arithmetic, algebra, and geometry.
In this exercise, we analyzed whether `x` is positive or negative using problem-solving skills specific to algebra. The step-by-step manner in which the equations were solved showcases how candidates should tackle GMAT quantitative questions:

• First, simplify and rearrange the given equations.
• Second, use algebraic techniques like factoring or multiplying to isolate the variable.
• Third, interpret the solutions to determine the answer to the question posed.

Developing these skills is crucial for success on the GMAT, as it enhances your ability to break down complex problems and find solutions efficiently.

Inequalities in Algebra

Inequalities in algebra involve expressions where the two sides are not necessarily equal, but have a relationship described by inequality symbols such as `>`, `<`, `≥`, `≤`. In this exercise, while we did not explicitly work with inequalities, understanding inequalities is necessary to grasp how different solutions can impact the interpretation of a problem.
For instance, when solving the equation \( \frac{4}{x} = x \), we derive two possibilities for `x` (i.e., \( x = 2 \) or \( x = -2 \)). These solutions imply that `x` could be positive or negative, highlighting why inequalities are fundamental in determining the range or possible values of variables.

• Analyzing inequalities helps us determine intervals where the expression holds true.
• Testing values within these intervals can verify if our solutions are consistent with the problem's conditions.

Mastering inequalities equips students with the analytical tools needed for higher-level math and quantitative reasoning tasks, ensuring they can accurately interpret and solve a variety of algebraic problems on the GMAT.