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Chapter 1: Problem 42

Let \(a, b,\) and \(c\) be any real numbers. Then \(a

Short Answer

Expert verified
We are given \(a+c<b+c\). By subtracting \(c\) from both sides, we get \(a<b\). We can express this inequality as \(a+x=b\) with \(x=b-a\), which is positive since \(a<b\). Thus, if \(a+c<b+c\), then \(a<b\).

Step by step solution

01

Rewrite the inequality

Given the inequality \(a+c<b+c\), we will start by rewriting it in terms of the given fact.
02

Find x

Subtract \(c\) from both sides of the inequality \(a+c<b+c\) to get a new inequality: \(a + c - c < b + c - c \Rightarrow a < b\) Now, we need to express this inequality in the form \(a+x=b\) for some positive real number \(x\). We can do this by finding the value of \(x\), which is the difference between \(b\) and \(a\), i.e. \(x = b - a\).
03

Show that x is positive

Since \(a < b\), then the difference between them, \(x = b - a\), must be a positive real number because \(b\) is greater than \(a\).
04

Conclusion

We've expressed the inequality \(a<c<b+c\) in the form \(a+x=b\) for some positive real number \(x\), which is the given fact. Thus, if \(a+c<b+c\), then \(a<b\).

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Key Concepts

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

Real Numbers
Real numbers are an invaluable part of mathematics encompassing all the numbers we are familiar with from everyday life: integers, fractions, and irrational numbers like \(\pi\) and \(\sqrt{2}\). They can be visualized as points on an endless number line.
Understanding real numbers is essential because they form the foundation for most mathematical concepts. In the context of inequalities, they allow us to discuss how values compare and interact.
  • Real numbers include positive numbers, negative numbers, and zero.
  • They can be both rational (like \(\frac{1}{2}, 4,\) and \(0.75\)) and irrational (like \(\sqrt{3}\), \(\pi\)).
Real numbers are crucial when expressing inequalities, as these expressions often require subtracting or adding values on the number line to establish their relationships.
Mathematical Proof
A mathematical proof is a logical argument that verifies the truth of a given statement or theorem. In our exercises involving inequalities, proofs help show that our logical deductions hold true consistently.
For inequalities, proofs often involve rearranging terms or properties like adding the same value to each side without flipping the inequality's direction.
  • Proofs can be done by direct arguments as we see in our problem; simplifying \(a + c < b + c\) to \(a < b\).
  • Using inequalities involves demonstrating properties such as transitivity, if \(a < b\) and \(b < c\), then \(a < c\).
When you establish a relation like \(a + x = b\) with \(x\) being a positive number, you convincingly show \(a < b\), adhering firmly to the inequality principle.
Positive Numbers
Positive numbers lie to the right of zero on the number line and are greater than zero. In mathematics, they play a critical role in verifying the truth of propositions involving inequalities.
Whenever you express an inequality such as \(a < b\) with \(a + x = b\) for a positive \(x\), it means that \(x\) is what shifts \(a\) up to \(b\).
  • Positive numbers are always used to increase values because their addition results in a value larger than the original.
  • The concept of positive numbers assures that when \(a + x = b\), we can deduce \(a < b\).
By recognizing that \(x = b - a\) is positive, you confirm \(a < b\), cementing a foundational concept in solving inequalities.

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