Question

Which of the following inequalities is the solution to the inequality 3x + 2 > 9x + 6? (A) \(x<-2 / 3\) (B) \(x>-2 / 3\) (C) \(x<-3 / 4\) (D) \(x>-3 / 4\) (E) \(x>2 / 3\)

Short answer

The solution is \( x < -\frac{2}{3} \).

Step-by-step solution

Set up the inequality

Start with the given inequality: \[ 3x + 2 > 9x + 6 \]

Move all x terms to one side

Subtract 3x from both sides to isolate the variable terms on one side: \[ 3x + 2 - 3x > 9x + 6 - 3x \]This simplifies to: \[ 2 > 6x + 6 \]

Move constant terms to the other side

Subtract 6 from both sides to isolate the term with the variable: \[ 2 - 6 > 6x + 6 - 6 \]This simplifies to: \[ -4 > 6x \]

Solve for x

Divide both sides by 6 to solve for x: \[ \frac{-4}{6} > x \]This simplifies to: \[ -\frac{2}{3} > x \]which is the same as: \[ x < -\frac{2}{3} \]

Key concepts

inequality solving

Inequality solving is a vital skill in mathematics and is frequently tested on exams like the GMAT. It involves finding the range of values that a variable can take to make an inequality true. The process includes:

• Identifying and isolating terms.
• Moving variable and constant terms to one side.
• Simplifying and solving for the variable.

For our example, we aim to solve the inequality:
\[ 3x + 2 > 9x + 6 \]
Following some carefully laid out steps, we shifted terms and simplified until we reached:
\[ x < -\frac{2}{3} \].
This step-by-step breakdown ensures that the inequality is both valid and comprehensible.

algebra

Algebra forms the backbone of much of GMAT math and involves manipulating equations to find unknown values. Here, we used fundamental algebra techniques to solve the inequality. Moving variables and constants involved:

• Subtracting 3x from both sides ->
  \[ 3x + 2 - 3x > 9x + 6 - 3x \],
  simplifying to
  \[ 2 > 6x + 6 \].
  
• Next, subtracting 6 from both sides ->
  \[ 2 - 6 > 6x + 6 - 6 \],
  simplifying to
  \[ -4 > 6x \].
  

These steps helped isolate x, and, finally, dividing both sides by 6 gave us:
\[ \frac{-4}{6} > x \], simplified to
\[ -\frac{2}{3} > x \] or
\[ x < -\frac{2}{3} \].
Understanding algebra is critical for solving such equations.

graduate management admission test preparation

When preparing for the GMAT, mastering inequality solving and algebra is crucial. This practice not only improves mathematical skills but also boosts analytical thinking. Here are some key tips:

• Practice regularly: The more you solve, the better you’ll get at identifying the steps needed.
• Learn to simplify: Simplifying equations step-by-step will reduce errors and save time.
• Use GMAT prep resources: Books, courses, and online platforms can provide additional practice problems and explanations.
• Stay calm and methodical: Break down each problem into manageable steps to avoid feeling overwhelmed.
• Review mistakes: Always check where you went wrong to understand and correct misconceptions.

By following these tips, and using resources like practice tests and tutoring, you’ll improve not only your math skills but also your performance on the GMAT.