Question

Which of the following is the set of all real numbers \(x\) such that \(x+3>x+5 ?\) A. The empty set B. The set containing all real numbers C. The set containing all negative real numbers D. The set containing all nonnegative real numbers E. The set containing only zero

Short answer

Answer: A. The empty set

Step-by-step solution

Simplify the inequality

We start by examining the inequality to see if we can simplify it. \(x+3 > x+5\)

Isolate the variable

Subtract \(x\) from both sides of the inequality to isolate the variable on one side: \(x - x + 3 > x-x + 5\) \(3 > 5\)

Evaluate the inequality

Since 3 is not greater than 5 (3 > 5 is false), the given inequality has no solution.

Choose the correct option

As there are no real numbers that would make the given inequality true, the solution corresponds to the empty set. So, the correct answer is: A. The empty set

Key concepts

Mathematical Inequalities

Mathematical inequalities are statements about the relative size or order of two values. They are an integral part of mathematics, particularly when dealing with the real number system. An inequality can typically be recognized by symbols such as (<, >, \(≤\), \(≥\), which stand for 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to', respectively. Unlike equations, inequalities do not imply equality but rather a range of potential values that satisfy the conditions of the inequality.

Understanding inequalities is crucial for solving various math problems, including those involving algebra, calculus, and even everyday decision-making. For example, when budgeting, you might need to ensure that your expenses are less than your income, an instance of a practical inequality.

Solving Inequalities

Solving inequalities involves finding the set of all real numbers that make the inequality true. The process is similar to solving equations, but instead of finding a single solution, we often find a range of solutions. Here are some common steps to solve an inequality:

• Isolate the variable on one side of the inequality.
• Simplify the inequality, if possible, by combining like terms and reducing fractions.
• Be mindful of the direction of the inequality sign, especially when multiplying or dividing by negative numbers, as this will reverse the inequality.
• Graphically represent the solution set on a number line, if needed, to visualize the range of possible solutions.

When solving the inequality from the exercise ((x+3 > x+5)), the process highlights that not all inequalities have a solution. In this case, the contradiction in the final step (3 > 5) revealed that the inequality is impossible to satisfy, leading to the correct answer of the empty set.

Real Number System

The real number system is the foundation on which the concept of inequalities rests. It consists of all the numbers on the number line, including rational numbers (like fractions and integers) and irrational numbers (like \(\sqrt{2}\) or \(\pi\)). Real numbers are used to represent continual values, such as distance, time, and temperature.

An important property of the real number system is its orderliness, meaning that any two real numbers can be compared, and one will always be greater, less, or equal to the other. This notion of order is what makes the use of inequalities possible. Furthermore, the real number system is complete; every point on the number line corresponds to a real number and vice versa. This completeness ensures there will always be a 'next' number, which is essential when discussing the solution sets of inequalities.