Problem 69 - Algebra for College Students

Question

Solve each problem by setting up and solving an appropriate inequality. How do you know by inspection that the solution set of the inequality \(x+3>x+2\) is the entire set of real numbers?

Step-by-Step Solution

Answer

The inequality holds for all real numbers since \(3 > 2\) is always true, independent of \(x\).

Step 1: Identify the Inequality

The inequality given is \(x + 3 > x + 2\).

Step 2: Simplify the Inequality

Subtract \(x\) from both sides of the inequality. The resulting inequality is \(3 > 2\).

Step 3: Interpret the Simplified Statement

Recognize that the inequality \(3 > 2\) is a true statement and does not depend on \(x\). Since the inequality holds true for any value of \(x\), the solution set includes all real numbers.

Key Concepts

Understanding Real Numbers

Real numbers encompass a wide range of numbers that we use in our day-to-day math. They include all the numbers we can think of: integers, fractions, and decimals. In more technical terms, real numbers consist of both rational numbers, like 1/2 or -6, and irrational numbers, such as \(\sqrt{2}\) and \(\pi\).
Real numbers can be positive, negative, or zero. They form a continuous number line that stretches infinitely in both directions.

• Rational Numbers: These numbers can be expressed as the quotient of two integers (like 1/3 or -7).
• Irrational Numbers: These cannot be written as a simple fraction (such as \(\sqrt{3}\) or \(\pi\)).
• Integers: These are whole numbers that can be positive, negative, or zero (like -3, 0, 7).

When solving inequalities, especially those involving the entire set of real numbers, it's crucial to understand this complete landscape.
For example, when you simplify an inequality to a true statement like \(3 > 2\), the solution applies to every real number because there are no restrictions binding it to only a subset of these numbers.

Basics of Mathematical Logic

Mathematical logic is like the law of the land in mathematics. It dictates the rules and processes we use to understand mathematical statements, and to determine their validity.
At its core, mathematical logic helps us to decide whether a statement is true or false, which is especially important in solving inequalities.

• Logical thinking involves clear reasoning and understanding how different mathematical statements interact.
• Predicate logic gives us tools to make universal or particular statements, like "For all \(x\), this is true" or "There exists an \(x\) such that".
• When we simplify inequalities to statements like \(3 > 2\), we're using logical proofs to establish these as universally true, independent of the variable \(x\).

In the inequality \(x + 3 > x + 2\), simplifying it to \(3 > 2\) is purely an exercise in following logical steps that prove the statement true for all 'x'. Therefore, it must hold for the entire set of real numbers.

Simplifying Inequalities

Simplifying inequalities is all about making them easier to understand. When we simplify, we aim to reduce them to a simpler or more recognizable form, often through basic algebraic manipulations. This sometimes involves removing terms that appear on both sides, as seen in the original problem.
To simplify the inequality \(x + 3 > x + 2\):

• Recognize common terms on both sides, here we have \(x\).
• Subtract \(x\) from both sides, leading us to the simpler inequality \(3 > 2\).

Once simplified, you can effectively assess the truth of the statement.
By reducing the given inequality to a universally true statement, \(3 > 2\), it is immediately clear that no particular value of \(x\) is needed to make this true. Therefore, the solution applies to all real numbers.
This simplification technique doesn't just save effort; it clarifies that the original inequality has no constraints or special conditions upon \(x\), other than being comfortably part of the real number set.