Question

Is y negative? (1) \(y^2-y-2=0\) (2) \(2 \mathrm{y}<0\) 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. 1 and 2 together are not sufficient

Short answer

B. 2 alone, not 1 alone

Step-by-step solution

Analyzing Statement (1)

Solve the quadratic equation given in Statement (1): \(y^2 - y - 2 = 0\)Factor the quadratic equation:\((y - 2)(y + 1) = 0\)Set each factor equal to zero:\(y - 2 = 0 \implies y = 2\)\(y + 1 = 0 \implies y = -1\)So, the solutions are: \(y = 2\) and \(y = -1\). The possible values for y from Statement (1) are 2 and -1. This alone is not sufficient to determine if y is negative.

Analyzing Statement (2)

Consider the inequality given in Statement (2): \(2y < 0\)Divide both sides by 2:\(y < 0\)This tells us directly that y is negative. This statement alone is sufficient to determine that y is negative.

Combining Statements (1) and (2)

Since Statement (2) alone is sufficient to determine that y is negative, we do not need to consider both statements together to determine whether y is negative or not.

Conclusion

From the analysis, it's clear that Statement (2) alone is sufficient to determine the answer.

Key concepts

GMAT math problem

The GMAT often includes questions designed to test your ability to understand and manipulate algebraic expressions. One such question type involves analyzing statements to determine specific properties of a variable. In this particular exercise, we need to determine if the variable \(y\) is negative by analyzing two given statements.

Understanding how to parse these statements and apply algebraic rules is crucial for success. Here, we have two different types of conditions: a quadratic equation and an inequality. We will break down each to find the solution.

Let's start by looking into the quadratic equation and its solutions. Then, we'll interpret the inequality and see how each statement contributes to the final answer.

quadratic equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). In this exercise, the quadratic equation given in Statement (1) is \(y^2 - y - 2 = 0\).

We solve quadratic equations by factoring, completing the square, or using the quadratic formula. For simplicity, we’ll focus on factoring in this exercise:

First, rewrite the equation as is and look for factors:

The given equation becomes \((y - 2)(y + 1) = 0\).

To find the solutions, set each factor equal to zero:

• \(y - 2 = 0\) implies \(y = 2\)
• \(y + 1 = 0\) implies \(y = -1\)

So, the solutions for \(y\) based on the quadratic equation are 2 and -1. This means \(y\) can be either a positive or a negative value.

Because one solution is positive and the other is negative, this statement alone isn't enough to determine if \(y\) is strictly negative.

inequalities in algebra

Inequalities are mathematical expressions involving the symbols \(<, >, \leq, \geq\). They show the relationship between two expressions. In Statement (2), the inequality provided is \(2y < 0\).

To solve the inequality, divide both sides by 2 to isolate \(y\):

\( \frac{2y}{2} < \frac{0}{2} \)
This simplifies to \(y < 0\).

The inequality \(y < 0\) directly tells us that \(y\) is negative. Consequently, this statement alone is sufficient to determine the nature of \(y\).

When solving GMAT math problems, understanding inequalities helps in quickly narrowing down the correct answer. They provide direct information about the ranges and limits of variables. In this case, Statement (2) gives us a clear indication that \(y\) is negative, making it sufficient on its own to answer the question.