Question

If \(a=x+2 y\), and \(b=y+2 x\), and \(3 x+7 y>7 x+3 y\), then which one of the following is true? (I) \(a>b\) (II) \(a=b\) (III) \(a

Short answer

(C) III only

Step-by-step solution

Analyze Given Inequality

We start by analyzing the given inequality \(3x + 7y > 7x + 3y\). Our goal is to simplify this inequality to understand how \(x\) and \(y\) relate.

Simplify the Inequality

Subtract \(7x\) and \(3y\) from both sides of the inequality: \(3x + 7y > 7x + 3y\). This becomes: \(-4x + 4y > 0\). Simplifying further, we get \(-x + y > 0\), which is equivalent to \(y > x\).

Compare Expressions for a and b

Substitute the values for \(a\) and \(b\): \(a = x + 2y\) and \(b = y + 2x\). We need to compare \(a\) and \(b\) based on the inequality \(y > x\).

Substitute y > x

From \(y > x\), rewrite \(a\) and \(b\) and compare them. We have: \(a = x + 2y\) and \(b = y + 2x\). Applying \(y > x\):- \(a < b\), because: \(x + 2y < y + 2x\) simplifies to \(y < x\), contradictory unless \(a < b\).So, our condition \(y > x\) supports \(a < b\).

Conclude the Configuration

Based on our understanding from comparisons, since \(y > x\) leads to \(a < b\), the correct option that holds true is (III) \(a < b\), which corresponds to option (C) III only.

Key concepts

Inequality Simplification

Inequality simplification is a critical skill in solving algebraic problems, especially on tests like the GMAT. It involves breaking down complex inequalities into simpler, equivalent forms to better understand the relationship between variables. This process often requires:

• Identifying like terms
• Performing arithmetic operations
• Isolating variables

For our exercise, we started with the inequality \(3x + 7y > 7x + 3y\). By subtracting \(7x\) and \(3y\) from both sides, the inequality simplifies to \(-4x + 4y > 0\). Further simplification leads to \(-x + y > 0\), which is equivalent to \(y > x\).
Through simplification, we've translated a multifaceted expression into an easier inequality, providing a clear insight into the hierarchy between \(x\) and \(y\). This simplification is crucial as it lays the groundwork for subsequent expression comparison.

Expression Comparison

When dealing with expressions, comparison revolves around evaluating and determining the relationships between different expressions based on given conditions. In the GMAT context, this often involves comparing expressions like \(a\) and \(b\).
From our exercise:

• \(a = x + 2y\)
• \(b = y + 2x\)

To compare these expressions, substitute the inequality \(y > x\) into the equations. Start by substituting and rearranging:

• For \(a = x + 2y\), consider the influence of \(2y\) given \(y > x\).
• For \(b = y + 2x\), note that \(2x\) is now smaller compared to \(y\)'s influence in \(a\).

After substitution and logical deduction, it makes sense that \(a < b\). This is due to the dominant presence of \(y + 2x\) outweighing \(x + 2y\) under the condition \(y > x\). Thus, expression comparison revealed \(a < b\), leading us to our conclusion.

GMAT Problem-Solving

GMAT problem-solving involves a structured method to tackle questions efficiently, focusing on logical reasoning and mathematical acumen. For inequalities and algebraic expressions, follow a clear process:

• Understand the problem and identify key variables.
• Simplify where possible to reveal relationships.
• Compare expressions logically under given constraints.
• Conclude by evaluating options against the simplified information.

In this specific problem, starting from an inequality simplification, and moving to expression comparison provided us the ability to logically deduce that \(a < b\). The successful solving of the problem rests on appropriately implementing methods to simplify, compare, and contrast algebraic expressions, typical in GMAT questions. This approach ensures accuracy and clarity, allowing for effective test performance.