Question

Is \(x>0\) ? (1) \(x^2-x=0\) (2) \(2 x^2-2 x=0\) 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

E. 1 and 2 together are not sufficient.

Step-by-step solution

- Analyze Statement 1

Start by analyzing the first statement: Given: (1) \[ x^2 - x = 0 \] Factorize the equation: \[ x(x-1) = 0 \] So, \[ x = 0 \text{ or } x = 1 \] This gives two possible values for x: 0 and 1. We need to determine if this statement is sufficient to confirm if \( x > 0 \). Only one of the solutions, x = 1, satisfies \( x > 0 \). However, since x = 0 is also a solution and does not satisfy \( x > 0 \), statement 1 alone is not sufficient.

- Analyze Statement 2

Next, analyze the second statement: Given: (2) \[ 2x^2 - 2x = 0 \] Divide the entire equation by 2: \[ x^2 - x = 0 \] This results in the same equation as statement 1: \[ x(x-1) = 0 \] Thus, x can be 0 or 1. Similar to statement 1, this is not sufficient to determine if x must be greater than 0, because x could also be 0.

- Combine Statements 1 and 2

Now, combine both statements: Both statements reduce to the equation \[ x(x-1) = 0 \] From this, x can be either 0 or 1. There is no additional information that restricts x to values greater than 0. The statements together still allow x to be 0, which fails to confirm \( x > 0 \).

- Conclusion

Neither statement 1 alone, statement 2 alone, nor both statements together can definitively prove that \( x > 0 \). Thus, the answer is that the statements are not sufficient to determine \( x > 0 \).

Key concepts

Inequalities

In this exercise, we are trying to determine if the value of \(x\) is greater than 0. This is an inequality problem because we are comparing \(x\) to the number 0 to see if it is on the positive side of the number line. Inequality problems often require us to test different ranges of values to see where a condition holds true. For example, common methods include:

• Adding and subtracting from both sides of the inequality.
• Multiplying or dividing both sides by a positive or negative number (keeping in mind that flipping the inequality sign is necessary when multiplying or dividing by a negative).
• Using graphical representation when appropriate.

In this specific problem, we are given conditions in the form of equations and are asked to see if they lead to \(x > 0\). Understanding the rules of inequalities is crucial when solving such problems.

Quadratic Equations

Quadratic equations form the backbone of this exercise. A typical quadratic equation is represented as \(ax^2 + bx + c = 0\). Here, we have factors of the equation in different forms. Let's dissect how we approached solving the quadratic equations:

• Factorizing: In the given problem, we started with \(x^2 - x = 0\). This was factorized into \(x(x-1) = 0\).
• Roots: Factoring allowed us to find the roots \(x = 0\) and \(x = 1\), which give us the solutions of the equation.
• Verifying Solutions: In both statements (1) and (2) in the exercise, we see that solving the quadratic equations provides potential values for \(x\). However, knowing these roots doesn't directly answer whether \(x > 0\).

Quadratic equations often come up in GMAT problem solving. They require careful manipulation and an understanding of how their roots affect the given condition or inequality.

Data Sufficiency Questions

Data sufficiency questions are a unique type of problem on the GMAT. They are designed to test your ability to determine whether provided information is enough to answer a question. Here, the question asked was whether \(x > 0\). We approached this in steps by evaluating the sufficiency of each given statement independently and then together.

• Evaluating Statements Independently: We analyzed statement (1) and concluded it was insufficient since it could lead to \(x = 0\) which fails the condition \(x > 0\). Then, we did the same for statement (2) which also led to an insufficient conclusion for the same reason.
• Combining Statements: We then combined statements (1) and (2). However, since both equations were essentially the same when simplified, combining them did not offer any new information, thus still leading to insufficient data to determine \(x > 0\).
• Answer Choices: Understanding how to choose the correct answer from the given options is essential. In this problem, neither statement alone nor together sufficed, leading us to conclude that the correct answer is (E).

Mastering data sufficiency questions requires practice and a logical approach to understand the provided data's limitations.