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Chapter 3: Problem 24

Which of the following inequalities is the solution to the inequality \(3 x-32.5\) (C) \(x>5 / 3\) (D) \(x<3 / 5\) (E) \(x<-1 / 2\)

Short Answer

Expert verified
The correct answer is (A) \(x < 2.5\)

Step by step solution

01

- Write down the inequality

Start by writing down the given inequality: \(3x - 3 < x + 2\)
02

- Move variables to one side

Subtract \(x\) from both sides to get all terms containing \(x\) on one side of the inequality: \(3x - x - 3 < 2\)
03

- Simplify

Combine like terms: \(2x - 3 < 2\)
04

- Isolate the variable

Add \(3\) to both sides to move the constant term to the other side: \(2x - 3 + 3 < 2 + 3\)which simplifies to\(2x < 5\)
05

- Solve for the variable

Divide both sides by \(2\) to isolate \(x\) \(x < \frac{5}{2}\)
06

- Choose the correct answer

Compare your result to the given options: Option (A) \(x < 2.5\) matches \(\frac{5}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving inequalities
Let's start by understanding what it means to solve inequalities. An inequality states that two expressions are not necessarily equal, and instead, one is greater than or less than the other. In this exercise, we are given the inequality \(3x - 3 < x + 2\). To solve it, we need to find the range of values for \(x\) that make this statement true. This involves several steps, such as simplifying the inequality, moving terms from one side to the other, and isolating the variable.
isolation of variables
Isolating the variable is a crucial step in solving inequalities and equations. In our example, after writing down the inequality \(3x - 3 < x + 2\), we moved all \(x\)-terms to one side by subtracting \(x\) from both sides, giving us \(2x - 3 < 2\). Following this, we isolate \(x\) by adding \(3\) to both sides to get \(2x < 5\). Finally, we divide both sides by \(2\) to solve for \(x\), yielding \(x < \frac{5}{2}\). Isolating \(x\) helps us clearly see the relationship between the variable and the constant.
mathematical reasoning
Mathematical reasoning is the logical process used to solve equations and inequalities. It involves a series of steps that follow logical rules and mathematical properties. In our problem, we used:
  • Subtraction Property of Inequalities, to subtract \(x\) on both sides.
  • Addition Property of Inequalities, to add \(3\) to both sides to isolate \(x\).
  • Division Property of Inequalities, to divide both sides by \(2\).
Each step maintains the balance of the inequality, ensuring the solution is correct.
GMAT preparation
Inequalities are a common part of the GMAT quantitative section. They test your ability to manipulate and solve for variables under specific conditions. To prepare well:
  • Practice solving a variety of inequalities to become familiar with different techniques.
  • Memorize properties of inequalities, like how multiplication or division by a negative number reverses the inequality sign.
  • Work on breaking down problems into smaller, manageable steps like our example above.
Being comfortable with these concepts will help you tackle inequality problems confidently on the GMAT.

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