Question

If \(2 y+x<0\) and \(y+2 x>0\) (A) \(-2 xy>-x / 2\) (C) \(2 x>y>-x / 2\) (D) \(-2 x

Short answer

The correct option is (A).

Step-by-step solution

Analyze the First Inequality

Starting with the inequality \(2y + x < 0\), isolate \(y\) by subtracting \(x\) from both sides, then divide by 2: \(2y + x < 0\) \(2y < -x\) \(y < -\frac{x}{2}\) Thus, \(y < -\frac{x}{2}\).

Analyze the Second Inequality

Next, consider the inequality \(y + 2x > 0\). Isolate \(y\) by subtracting \(2x\) from both sides: \( y + 2x > 0\) \( y > -2x \) Thus, \(y > -2x\).

Combine the Results

Combine the results from the two inequalities: From Step 1: \(y < -\frac{x}{2}\) From Step 2: \(y > -2x\) We get the combined inequality: \( -2x < y < -\frac{x}{2} \).

Identify the Correct Option

The correct option that matches our combined inequality \( -2x < y < -\frac{x}{2} \) is (A) \(-2 x<y<-x / 2\).

Key concepts

GMAT preparation

Preparing for the GMAT involves practicing various types of mathematical problems, including those that test your understanding of inequalities. Inequalities are common in GMAT problems and require a solid grasp of comparing expressions and understanding their implications.

In GMAT preparation, the goal is to manage time effectively while solving problems accurately. Practicing inequalities helps develop skills in logical reasoning and algebraic manipulation. Here are some useful tips:

• Familiarize yourself with key concepts in inequalities, such as properties of inequalities and rules for manipulating them.
• Understand how to isolate variables to simplify inequality expressions.
• Practice combining multiple inequalities to find overlapping solution ranges.

By incorporating these tips into your study routine, you'll be better equipped to handle inequalities on the GMAT.

inequalities

Inequalities describe the relationship between two expressions that are not equal. They are crucial in understanding the range of possible values for variables in mathematical problems. There are a few primary concepts you need to master:

First, it's important to know how to solve simple inequalities. For example, given the inequality \(2y + x < 0\), you need to isolate 'y': Subtract 'x' from both sides to get \(2y < -x\), and then divide by 2 to find \(y < -\frac{x}{2}\).

Next, learn to handle compound inequalities. Consider two separate inequalities: \(2y + x < 0\) becomes \(y < -\frac{x}{2}\) and \(y + 2x > 0\) becomes \(y > -2x\). Combining these, you get the inequality \( -2x < y < -\frac{x}{2}\).

Finally, understand how to check the solutions against provided options to identify the correct answer. In our example, we saw that the combined inequality \( -2 x < y < -\frac{x}{2}\) matches answer choice (A).

mathematics problem solving

Solving mathematical problems, particularly those involving inequalities, requires a systematic approach. Here are some key steps to keep in mind:

First, analyze each inequality separately to understand the constraints it places on the variables. For instance, isolating 'y' in \(2y + x < 0\) or \(y + 2x > 0\) requires careful manipulation of terms.

Second, draw a number line or use visual aids to help understand the solution set and how different inequalities intersect or overlap.

Third, combine the inequalities by finding the common solution range. As in our example, after isolating and transforming each inequality, you get a combined range that helps in identifying the possible values for 'y'.

Finally, review all options, if given, to ensure that the combined inequality matches one of the provided choices. This methodical approach helps in avoiding mistakes and finding the most accurate solution efficiently.