Question

Which of the following inequalities is the solution to the inequality \(x+3>x+2\) ? A. \(x>0\) B. \(x<0\) C. \(x<-\) infinity D. \(x>-\) infinity E. Cannot be determined

Short answer

D. \(x > -\infty\)

Step-by-step solution

Understand the Given Inequality

The given inequality is: \[x + 3 > x + 2\] We need to find the solution to this inequality.

Simplify the Inequality

Subtract \(x\) from both sides of the inequality:\[x + 3 - x > x + 2 - x\]This simplifies to:\[3 > 2\]

Analyze the Result

The inequality \[3 > 2\] is always true for all values of \(x\). Therefore, the inequality \(x + 3 > x + 2\) does not impose any restriction on \(x\).

Interpret the Options

Given that the inequality holds true for all \(x\):- Option A \(x > 0\) is not correct because the inequality is true for all values of \(x\).- Option B \(x < 0\) is not correct for the same reason.- Option C \(x < -\infty\) is not correct as it contradicts the fact that the inequality holds for all \(x\).- Option D \(x > -\infty\) is the correct interpretation because it means all values of \(x\).- Option E \(Cannot be determined\) is not correct since we can determine the solution.

Key concepts

Inequalities

Inequalities are mathematical expressions that compare two values using symbols like >, <, \geq, and \leq. In this exercise, we had to solve the inequality \((x + 3 > x + 2)\).

Let's break down the steps:

First, we simplified the inequality by subtracting x from both sides. This step removed x from the equation: \((x + 3 - x > x + 2 - x)\). We got \((3 > 2)\), which is true for any value of x.

This means there are no specific restrictions on x, and it can be any number. When solving inequalities, always simplify and then analyze what the simplification implies about the variable.

GMAT Preparation

Preparing for the GMAT involves mastering various mathematical concepts, including inequalities. Inequalities are a common topic on the GMAT, and you should be familiar with solving, simplifying, and interpreting them.

Key tips for GMAT preparation with inequalities:

• Always first simplify the inequality if possible.
• Look for operations that can isolate the variable.
• Analyze the simplified inequality to determine what it means for all possible values of the variable.

In this problem, identifying that \((3 > 2)\) is always true helped us understand that the inequality holds for any x. Thus, we concluded that our solution set includes all values of x.

Mathematical Reasoning

Mathematical reasoning involves understanding and finding relationships between quantities and making logical conclusions. When solving inequalities, reasoning is crucial.

Here’s how to apply mathematical reasoning to our problem:

• First, recognize the type of inequality and the operations allowed (e.g., adding/subtracting terms on both sides).
• Simplify the inequality step by step.
• Once simplified, infer the potential range of values for the variable that satisfy the inequality.

In the given problem, we simplified \((x + 3 > x + 2)\) to \((3 > 2)\). Realizing that this statement is universally true provided clarity: x can be any number. Thus, we deduced that the correct option is the one that represents all possible values for x. This kind of logical reasoning is essential for successfully tackling GMAT questions.