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Chapter 12: Problem 271

If \(1<\mathrm{a}\), show that \(\mathrm{a}<\mathrm{a}^{2}\).

Short Answer

Expert verified
Given that \(1 < a\), we square the inequality to obtain \(a^2 > 1\). Comparing the original inequality (\(a > 1\)) to the squared inequality, we can combine them to get \(1 < a < a^2\), which proves that if \(1 < a\), then \(a < a^2\).

Step by step solution

01

State the given condition

We are given that \(1 < a\), which means \(a\) is a positive number greater than 1.
02

Square the given inequality

Since \(a > 1\), we square both sides of the inequality to get: \[a^2 > 1^2\] This simplifies to: \[a^2 > 1\]
03

Compare the original inequality to the squared inequality

Now, let's compare the two inequalities: - The original inequality is \(1 < a\), which is equivalent to \(a > 1\). - The squared inequality is \(a^2 > 1\).
04

Combine the inequalities

Since \(a\) is a positive number greater than 1, and \(a^2\) is greater than 1, we can combine the inequalities to get: \[1 < a < a^2\] This proves that if \(1 < a\), then \(a < a^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Squaring Inequalities
When working with inequalities in algebra, squaring both sides is a common step that can help us derive new expressions. However, you have to be cautious, as this move can alter the inequality if not done under the right conditions. For instance, squaring any positive number retains the inequality direction, that is, if one side is greater than the other, squaring keeps it that way.
In the given exercise, since we start with the fact that \(1 < a\), our value of \(a\) is greater than 1, which ensures it's positive. Squaring gives us \(a^2\) and \(1^2\), or simply \(a^2>1\), which logically follows from our original inequality of \(a>1\).
  • If \(a>1\), then \(a^2>1\).
  • Squaring maintains the relationship as long as both numbers are positive.
Positive Numbers
Understanding the nature of positive numbers is crucial when discussing inequalities. Positive numbers are those that are greater than zero. In the context of inequalities, working with positive numbers often simplifies operations such as squaring or multiplying both sides of an inequality.
For the inequality \(1 < a\), knowing that \(a\) is greater than 1 confirms it is a positive number. This is important because multiplying or squaring positive numbers keeps the inequality direction unchanged.
  • Operations like squaring won't flip the inequality with positive numbers.
  • Positive numbers ensure consistency when modifying inequalities.
Algebraic Proof
Algebraic proofs are a formal method of demonstrating that a statement follows logically from given premises. In algebra, an algebraic proof typically involves a series of steps, each justified by algebraic properties or previously established results.
In proving that \(1 1\) and establishing \(a^2 > 1\) by squaring both sides. This step leads us to the conclusion that \(a
  • Algebraic proofs use logical reasoning to establish truth.
  • Each step in the proof is backed by previous results or algebraic laws.
Inequality Properties
In algebra, inequalities come with a set of guidelines and properties that are crucial for manipulation. Inequalities, like equations, can be manipulated to find a relation between different expressions. Some key properties to be aware of include:
  • If you add or subtract the same value from both sides of an inequality, its direction remains unchanged.
  • Multiplying or dividing by a positive number does not change the inequality's direction.
  • Squaring squishable positive numbers just like multiplying, holds the inequality true.
In our exercise, recognizing \(1 < a\) and seeing that it leads to \(a < a^2\) is possible by understanding these foundational properties of inequality.

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Most popular questions from this chapter

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

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