Question

Which of the following inequalities is the solution to the inequality \(x / 21 / 3\) B. \(x<1 / 6\) C. \(x<-2 / 3\) D. \(x>-1\) E. \(\mathrm{x}<0\)

Short answer

E. \( x < 0 \)

Step-by-step solution

Understand the Inequality

Given the inequality \( \frac{x}{2} < \frac{x}{3} \), we need to determine when it holds true.

Eliminate the Fractions

To make the inequality simpler, eliminate the fractions by finding a common denominator. Multiply both sides by 6 (the least common multiple of 2 and 3): \[ \frac{x}{2} \times 6 < \frac{x}{3} \times 6 \]

Simplify the Expression

Simplify the left-hand side and the right-hand side: \[ 3x < 2x \]

Isolate the Variable

Subtract 2x from both sides to isolate the variable x: \[ 3x - 2x < 0 \]

Solve for x

Simplify the inequality: \[ x < 0 \]

Choose the Correct Answer

The inequality x < 0 corresponds to choice E. Therefore, the solution is E.

Key concepts

solving inequalities

Solving inequalities is a core concept in algebra and essential for the GMAT. The aim is to find all possible values of the variable that make the inequality true. For example, consider the inequality given in the problem \(\frac{x}{2} < \frac{x}{3}\). The first step is to understand the inequality and determine which values of x satisfy this condition. Much like solving equations, inequalities involve similar steps, such as combining like terms, using inverse operations, and simplifying.

However, there are unique rules to follow:

• If you multiply or divide by a negative number, the inequality sign flips.
• Subtraction and addition do not change the direction of the inequality.

Remember, solving inequalities requires careful manipulation of the inequality sign, ensuring that each step maintains the truth of the original inequality.

common denominators

Finding a common denominator is crucial when dealing with fractions in inequalities. It means bringing different fractions to a comparable base, facilitating simpler calculations. In the given problem, \(\frac{x}{2} < \frac{x}{3}\), the common denominator is 6, the least common multiple of 2 and 3.

To eliminate the fractions, multiply both sides by this common denominator:

\[ \frac{x}{2} \times 6 < \frac{x}{3} \times 6 \]

This gives you:

\[3x < 2x \]

With the fractions eliminated, the inequality is easier to manage. Common denominators are a powerful tool to simplify and solve problems involving multiple fractions, enhancing clarity in your calculations and reducing the likelihood of errors.

isolating variables

Isolating the variable is a fundamental algebraic technique. It involves manipulating the equation or inequality to have the variable on one side and everything else on the other. In the problem given, after eliminating fractions and simplifying, you get:

\[3x < 2x\]

To isolate x, subtract 2x from both sides:

\[3x - 2x < 0\]

Simplifying further, you obtain:

\[x < 0\]

This inequality tells you the range of values for x that satisfy the original inequality. Isolating the variable helps to clearly identify the solution set and ensures accuracy.

critical reasoning

Critical reasoning goes beyond just solving the inequality—it involves interpreting the results and understanding their implications. In this problem, once you find that \(\frac{x}{2} < \frac{x}{3}\) simplifies to \(x < 0\), you must then carefully consider the given multiple-choice options. The correct answer is the one that matches your solution.

Always take time to analyze the results:

• Ensure the solution fits all conditions of the original problem.
• Beware of common traps or misinterpretations.
• Re-examine your steps if the options seem nonsensical.

Effective critical reasoning ensures robust and correct solutions to inequality problems, aligning mathematical solutions with logical and conceptual understanding.