Question

Which of the following inequalities is the solution to the inequality \(3 x+32\) C. \(x<-2\) D. \(x<-1\) E. No solution exists

Short answer

D. \(x < -1\)

Step-by-step solution

- Subtract x from both sides

Start by isolating the variable on one side. Subtract x from both sides of the inequality: \[ 3x + 3 - x < x + 1 - x \] This simplifies to: \[ 2x + 3 < 1 \]

- Subtract 3 from both sides

Subtract 3 from both sides of the inequality to further isolate the term with the variable: \[ 2x + 3 - 3 < 1 - 3 \] This simplifies to: \[ 2x < -2 \]

- Divide both sides by 2

Divide both sides of the inequality by 2 to solve for x: \[ \frac{2x}{2} < \frac{-2}{2} \] This simplifies to: \[ x < -1 \]

Key concepts

solving inequalities

Inequalities are similar to equations but instead of an equals sign, they use inequality signs like <, >, ≤, and ≥. To solve inequalities, we often use algebraic manipulation to isolate the variable.
For example, in the exercise given, we start with the inequality: \(3x+3We follow specific steps to isolate the variable:

• Subtract x from both sides: \(3x+3-x
• Subtract 3 from both sides: \(2x+3-3<1-3\), which simplifies to: \(2x<-2\).
• Divide both sides by 2: \(\frac{2x}{2}< \frac{-2}{2}\), which simplifies to: \(x<-1 \).

Each step involves an arithmetic operation that helps to isolate x and find the solution to the inequality.

GMAT preparation

Preparing for the GMAT involves honing your skills in various areas, including the ability to solve inequalities. This concept is essential because GMAT questions often test your understanding of algebra and your ability to perform algebraic manipulations efficiently.
Here are some tips for preparing for the GMAT:

• Practice regularly: The more you practice solving inequalities, the more comfortable you will become.
• Understand each step: Make sure you grasp why each algebraic manipulation is performed.
• Time management: Ensure you can solve these problems quickly and accurately, as time management is crucial in the GMAT.

By following these tips, you will improve your ability to solve inequalities and your overall GMAT performance.

algebraic manipulation

Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions to solve equations and inequalities.
In the exercise, the following algebraic manipulations were applied:

• Subtracting a term from both sides: This is done to simplify the inequality and isolate the variable. For example, subtracting x from both sides in the original equation.
• Subtracting a constant: This helps in further isolating the variable term. In the exercise, we subtracted 3 from both sides to simplify the inequality.
• Dividing by a coefficient: This is the final step to isolate x. For instance, dividing both sides by 2.

These manipulations ensure that we can solve for the variable step by step and arrive at the correct solution: \(x<-1\). Understanding these operations is key to mastering algebra and solving GMAT questions effectively.