Problem 47 - Prealgebra

Question

CHALLENGE Is it always, sometimes, or never true that \(x-1

Step-by-Step Solution

Answer

The inequality \(x-1 < x\) is always true for all real numbers, \(x\).

Step 1: Understanding the Inequality

We start with the inequality \(x - 1 < x\). This inequality asks if subtracting 1 from any number \(x\) results in a value less than the original number \(x\).

Step 2: Solving for x

Let's assume \(x - 1 = y\). If we rearrange this, we find that \(x = y + 1\). Substituting this back into the inequality, we get \(y < y + 1\).

Step 3: Analyzing y < y + 1

The inequality \(y < y + 1\) holds true for all real numbers. Since 1 is always added to \(y\), \(y\) will always be less than \(y + 1\).

Step 4: Conclusion

Since the rearranged inequality \(y < y + 1\) holds for each real number, we determine that the original statement \(x - 1 < x\) is always true for all \(x\).

Key Concepts

Prealgebra

When starting out with prealgebra, understanding inequalities is a vital skill. An inequality is simply a mathematical expression that shows the relationship between two values that are not equal.
In this particular exercise, the inequality is represented as \(x - 1 < x\), which might initially seem intricate, but it's quite straightforward.

To decode this, think of prealgebra as a friendly introduction to algebra, focusing on basic operations like addition, subtraction, and understanding inequalities.

• Inequalities involve symbols such as \(<\), \(>\), \(\leq\), and \(\geq\).
• They express one value being less than, greater than or equal to another.
• Here, we have \(x - 1\), which implies that you are taking 1 away from \(x\).
• The inequality \(x - 1 < x\) tells us that the result of this subtraction is always less than \(x\).

By fully grasping the basic notion that subtracting from a number naturally makes it smaller, you strengthen your prealgebra foundation.

Real Numbers

The set of real numbers encompasses all the numbers you use in everyday calculations and measurements. Real numbers include a vast array of quantities such as integers, fractions, and irrationals like \(\sqrt{2}\).

Understanding real numbers is crucial in evaluating inequalities like \(x - 1 < x\).

• Real numbers are continuous, meaning there are no gaps. Between any two real numbers, there's always another real number.
• They can be positive, negative or zero.
• Since the inequality in question, \(x - 1 < x\), holds true for all real numbers, it underscores the fundamental nature of numbers: subtracting 1 from \(x\) will always yield a value less than \(x\).

This characteristic of real numbers explains why the inequality is always true across their entire span. Thus, understanding real numbers allows you to evaluate such inequalities with confidence.

Mathematical Reasoning

Mathematical reasoning is all about making sense of numbers and their relationships using logic and deductive thinking.
In the context of inequalities like \(x - 1 < x\), reasoning helps in identifying why a statement holds true or not across different scenarios.

Here's how logical thinking applies:

• You start by asking what happens when 1 is subtracted from any real number. Clearly, the number gets smaller.
• By rearranging the equation, like in the solution \(y < y + 1\), we apply logic to see it must be true for any variable \(y\).
• This logical step solidifies the conclusion: for any real value of \(x\), the original statement \(x - 1 < x\) is universally true.

Developing robust mathematical reasoning skills ensures that you can handle inequalities and other algebraic expressions with ease. It encourages you to think critically and solve problems efficiently, influencing all areas of mathematics and beyond.