Question

Which of the following inequalities is the solution to the inequality \(2 x+1>x+2\) ? A. \(x>0\) B. \(x>2\) C. \(x>1 / 2\) Answer D. \(x>1\) E. Cannot be determined

Short answer

The correct inequality is \(x > 1\), which corresponds to option D.

Step-by-step solution

- Subtracting x from both sides

Start by subtracting x from both sides of the inequality: \(2x + 1 > x + 2\)Subtract x: \(2x + 1 - x > x + 2 - x\)Simplified to: \(x + 1 > 2\)

- Subtracting 1 from both sides

Next, subtract 1 from both sides of the inequality: \(x + 1 > 2\)Subtract 1: \(x + 1 - 1 > 2 - 1\)Simplified to: \(x > 1\)

- Identifying the correct option

Compare the simplified inequality \(x > 1\) with the options provided to find the correct answer, which is option D.

Key concepts

Solving Inequalities

Solving inequalities is a core component of algebra and is frequently tested in exams like the GMAT. Inequalities express a relationship of comparison between two mathematical expressions. The goal is to isolate the variable on one side of the inequality to find the range of values that satisfy it.

In the given problem, the inequality posed is: \[\begin{equation} \begin{aligned} &2x + 1 > x + 2 \ \text{By subtracting } x \text{ from both sides, we get:} \ &2x + 1 - x > x + 2 - x \ &x + 1 > 2 \ \text{Next, subtract 1 from both sides:} \ & x + 1 - 1 > 2 - 1 \ &x > 1 \ \text{Thus, the solution to the inequality is } x > 1. \ \text{This is option D in the given multiple-choice options.} \ \ \text{Always remember the importance of maintaining the direction of the inequality when performing operations. If we multiply or divide both sides by a negative number, the inequality symbol must be flipped.} \end{aligned} \ \end{equation}\].

Graduate Management Admission Test (GMAT)

The GMAT is a standardized exam used by business schools worldwide to assess candidates for graduate-level management programs. The test measures a range of skills, with a focus on analytical writing, quantitative problem solving, and verbal reasoning.

Success in the GMAT demands strong mathematical reasoning skills. Many questions involve algebraic concepts, like solving inequalities, which test a student's ability to manipulate and analyze equations efficiently.

To prepare effectively for the GMAT:

• Practice consistently with different types of problems.
• Develop a clear understanding of key mathematical concepts, such as algebra, geometry, and arithmetic.
• Hone your test-taking strategies, including time management and elimination methods.
• Utilize practice exams to simulate test conditions and identify areas for improvement.

Being familiar with inequalities and other algebraic operations is imperative for doing well in the quantitative section of the GMAT.

Mathematical Reasoning

Mathematical reasoning is the ability to think logically and use mathematical concepts to solve problems. It is a critical skill that involves understanding, analyzing, and working through various mathematical challenges.

When faced with an inequality on the GMAT:

• Identify the operations required to isolate the variable.
• Maintain the inequality's direction through each operation.
• Understand that multiplying or dividing both sides by a negative number requires flipping the inequality sign.
• Always verify the solution by substituting values back into the original inequality to ensure it holds true.

Logical steps and careful execution are essential for efficient problem solving. Practice helps build intuition and speed, which are invaluable during timed exams like the GMAT.