Question

Prove that if \(\mathrm{a}>\mathrm{b}>0\), then \(1 / \mathrm{a}<1 / \mathrm{b}\)

Short answer

Given the inequality \(\mathrm{a}>\mathrm{b}>0\), we divide both sides by \(\mathrm{a}\mathrm{b}\) to get \[\frac{\mathrm{a}}{\mathrm{a}\mathrm{b}}>\frac{\mathrm{b}}{\mathrm{a}\mathrm{b}}\]. After simplifying, we are left with the inequality \(\frac{1}{\mathrm{b}}>\frac{1}{\mathrm{a}}\), which can be rewritten as \(1/\mathrm{a}<1/\mathrm{b}\), proving the statement.

Step-by-step solution

Statement of Inequality

We are given the inequality \(\mathrm{a}>\mathrm{b}>0\).

Manipulate the Inequality

We'll manipulate the given inequality so that it eventually shows us the desired result. Since both \(\mathrm{a}\) and \(\mathrm{b}\) are positive, we can divide both sides of the inequality by the product \(\mathrm{a}\mathrm{b}\) without changing the direction of the inequality. So, we now have: \[\frac{\mathrm{a}}{\mathrm{a}\mathrm{b}}>\frac{\mathrm{b}}{\mathrm{a}\mathrm{b}}\]

Simplify the Inequality

Now, we can simplify both sides of the inequality by cancelling out the common terms: On the left-hand side, the \(\mathrm{a}\) terms cancel out, giving us: \[\frac{1}{\mathrm{b}}\] On the right-hand side, the \(\mathrm{b}\) terms cancel out, giving us: \[\frac{1}{\mathrm{a}}\] So, we are left with: \[\frac{1}{\mathrm{b}}>\frac{1}{\mathrm{a}}\]

Conclusion

The inequality \(\frac{1}{\mathrm{b}}>\frac{1}{\mathrm{a}}\) is equivalent to \(1/\mathrm{a}<1/\mathrm{b}\). Therefore, we have shown that if \(\mathrm{a}>\mathrm{b}>0\), then \(1/\mathrm{a}<1/\mathrm{b}\).

Key concepts

Manipulating Inequalities

When working with inequalities, we often have to manipulate them in order to make comparisons or find solutions. The key rule to remember is that the direction of the inequality doesn't change when both sides are multiplied or divided by a positive number. However, if we multiply or divide by a negative number, the inequality flips.

In the given exercise, we divide both sides of the original inequality \(a > b\) by the positive product \(ab\), maintaining the inequality direction. Always ensure that the numbers you're using to manipulate the inequality do not change sign within the range you're working with. This is why it was crucial that both \(a\) and \(b\) are greater than zero, ensuring that their product is also positive, allowing us to safely divide without flipping the inequality sign.

Positive Number Properties

Positive numbers have special properties that are very useful when solving inequalities. One of these is that the product of two positive numbers is always positive. This comes into play in our problem since \(a\) and \(b\) are both greater than zero, their product \(ab\) is also positive.

Another important property is that if you have two positive numbers, the greater number will have a smaller reciprocal than the smaller number. That's precisely what we're trying to prove here: since \(a > b\), we can predict that \(1/a < 1/b\) based solely on understanding the nature of positive numbers. This intuitive understanding is backed up by methodical proof.

Simplifying Algebraic Expressions

Simplification is a process of rewriting an expression in a more accessible and understandable way. It often involves reducing fractions, removing parentheses, or canceling out common factors. In our inequality problem, simplification is used to demonstrate the relationship between the two sides of the inequality after they have been divided by the positive number \(ab\).

This step dramatically simplifies the problem and brings us closer to the answer. By canceling out \(a\) on the left side and \(b\) on the right side, we're able to see that \(1/b > 1/a\), thus achieving the simplicity needed to prove our point. Remember, simplifying algebraic expressions doesn't alter their value or the inequality's truth; it just makes it easier to grasp and work with.

Logical Reasoning in Mathematics

Mathematics isn't just about computations; it's also about using logic to form conclusions. In inequality proofs, logical reasoning helps us transition from one statement to the next in a coherent and justified manner. In our problem, the proof starts with the assumption \(a > b > 0\) and logically leads to the conclusion \(1/a < 1/b\) through a series of justifiable steps.

Through logical reasoning, we establish that since \(a\) and \(b\) are both positive and \(a\) is greater than \(b\), then dividing by a larger number \(a\) should give us a smaller result than dividing by a smaller number \(b\). It's this type of deductive thinking that allows us to carry out each step of the proof with confidence that we're moving in the right direction towards proving what we have set out to prove.