Problem 81 - Elementary Algebra Within Reach

Question

Determine whether the statement is true or false. Justify your answer. $$x

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understanding the Given Inequality

We are given an inequality \(x < x + 4\). This inequality compares \(x\) to \(x + 4\).

Step 2: Manipulate the Inequality

If we subtract \(x\) from both sides of the inequality to isolate the variable on one side, the inequality becomes 0 < 4.

Step 3: Evaluate the Result

On the number line, 0 is indeed less than 4. So, the inequality is true.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions used to compare two values or expressions. They show the relationship of one value being greater than, less than, greater than or equal to, or less than or equal to another. Think of them as directions that tell you how one quantity relates to another.
In an inequality such as \(x < x + 4\), it indicates that the value of \(x\) is less than \(x + 4\). An important part of understanding inequalities is recognizing the symbols involved:

• \(<\) means "is less than."
• \(>\) means "is greater than."
• \(\leq\) indicates "is less than or equal to."
• \(\geq\) indicates "is greater than or equal to."

Recognizing these symbols helps you understand the relationship between the expressions involved.

Mastering Inequality Manipulation

Inequality manipulation is a critical skill because it allows you to reshape and understand inequalities more clearly. It involves applying legal operations to both sides of an inequality without changing the relative order of the expressions.
In our example, the initial inequality is \(x < x + 4\). We want to see clearly how this inequality holds by rearranging it. A common technique is to subtract the same expression from both sides. This ensures that the balance of the inequality remains unaltered.
For \(x < x + 4\), when we subtract \(x\) from both sides, we simplify it to \(0 < 4\). By reducing it down to this basic inequality, it's evident that 0 is indeed less than 4, verifying the truth of the original statement. This manipulation involves:

• Subtracting or adding the same value on both sides.
• Multiplying or dividing both sides by a positive number.
• Remember! When multiplying or dividing both sides by a negative number, the inequality sign flips direction.

By mastering these techniques, you can solve and understand a vast range of inequality problems.

Exploring Basic Algebra Concepts

Basic algebra is the foundation of most mathematical learning. It involves understanding how to manipulate numbers and variables to find unknowns and see how different elements interact within equations and expressions.
Algebra introduces the use of variables like \(x\) to symbolize unknowns. In our problem-solving process, we see basic algebra at work when we isolate terms to simplify expressions, such as moving \(x\) to one side in the inequality \(x < x + 4\).

• Isolating variables involves operations like addition, subtraction, multiplication, and division.
• It's essential to maintain equality or the inequality through these operations.
• Basic algebra helps in building the understanding required for more involved mathematical concepts later on.

By engaging with simple algebra concepts, students develop skills that are transferable and critical for tackling more complex mathematical and real-world problems.