Problem 56 - Calculus: One and Several Variables

Question

Let \(a . b . c\) be nonnegative numbers. Show that $$\text { if } \quad a \leq b+c, \quad \text { then } \quad \frac{a}{1+a} \leq \frac{b}{1+b}+\frac{c}{1+c}.$$

Step-by-Step Solution

Answer

Given the condition that \(a \leq b+c\), we have transformed and simplified the inequality to \((b-c)(a-c) \geq 0\). This condition will always hold true, proving the initial inequality \(\frac{a}{1+a} \leq \frac{b}{1+b}+\frac{c}{1+c}\).

Step 1: Rewriting the inequality

First, rewrite the inequality to bring it to a simpler form that allows easier comparison. To achieve this, we can use subtraction to remove some terms from the right side. This results in \(\frac{a}{1+a} - \frac{b}{1+b} - \frac{c}{1+c}\). Then the inequality becomes \(\frac{a}{1+a} - \frac{b}{1+b} - \frac{c}{1+c} \leq 0\).

Step 2: Simplifying the inequality

Next, we multiply by \((1+a)(1+b)(1+c)\) to clear the denominators and expand the expressions. This will yield the inequality \(ac(1+b)+bc(1+a) \leq bc(1+a)+ac(1+b)+abc\).

Step 3: Rearranging and factoring the inequality

Rearrange the terms and then factor the resulting polynomial inequality so that it becomes \((b-c)(a-c) \geq 0\). This way the inequality is simplified.

Step 4: Checking if the condition passes the inequality

In order to check if the condition \(a \leq b+c\) passes the inequality \((b-c)(a-c) \geq 0\), it helps to realize that since \(a \leq b+c\), then \(a-c \leq b\), which makes \((b-c)(a-c) \geq 0\) for all nonnegative a, b, and c.

Key Concepts

Nonnegative Numbers

Nonnegative numbers are a fundamental concept in mathematics. A nonnegative number is any number that is greater than or equal to zero. This means it can be a positive number or zero itself. When specified in mathematical problems, especially inequalities, it is crucial to ensure all variables are nonnegative to maintain the truth of the statement.
For example, in the given inequality problem with variables a, b, and c, each of these is nonnegative.

• This impacts the inequality as it ensures no fractions become negative.
• Nonnegative numbers help establish a domain where certain mathematical proofs, such as inequalities involving fractions or products, hold true.

Recognizing the nonnegative condition allows you to simplify and manipulate the inequality confidently, knowing the outcome remains valid over the specified range.

Simplifying Inequalities

Simplifying inequalities is a critical skill in mathematics that helps solve or prove inequalities. The main goal is to transform the given inequality into a simpler form that either shows the relationship more clearly or is easier to work with for further manipulation or proof.
In our problem, the original inequality is transformed by first rewriting it as \[\frac{a}{1+a} - \frac{b}{1+b} - \frac{c}{1+c} \leq 0\]This step helps to consolidate terms for comparison.

• Next, clearing the denominators by multiplying both sides by \((1+a)(1+b)(1+c)\) further simplifies the inequality.
• This removes fractions and provides a polynomial form that is often easier to manipulate.

Simplifying an inequality involves clearing fractions, factoring, and rearranging terms, making it easier to test the conditions provided in the problem statement.

Polynomial Inequalities

Polynomial inequalities involve expressions made up of polynomials and their inequalities often require careful rearrangement and factoring. In our problem, after simplifying the fraction terms, the inequality transforms into a polynomial form, \[ (b-c)(a-c) \geq 0 \]
This polynomial inequality is key to solving the problem.

• The expression \((b-c)(a-c)\) can be interpreted as a product of factors, which can be analyzed based on the signs and values of b and c.
• Understanding how to factor polynomials and the impact of each factor's sign provides insight into how the inequality holds true or not.

Working with polynomial inequalities might involve identifying points where the inequality changes from true to false or vice versa, using methods like test points or analyzing vertex points in a quadratic scenario.

Mathematical Proofs

Proving mathematical statements requires a logical sequence of steps that demonstrate why a statement is universally true. Proofs are a backbone of mathematics, offering a rigorous explanation that stands under scrutiny.
The provided inequality proof involves verifying the relationship under given conditions. Here’s what this involves:

• Firstly, start by transforming the inequality into a form that is easier to handle and interpret.
• Secondly, analyze the transformed inequality using known mathematical properties, such as nonnegative values and their impact on inequality signs.
• Thirdly, you logically conclude by linking each step to prove that under the given condition \(a \leq b+c\), the inequality holds.

This type of proof, often called a direct proof, systematically demonstrates how the initial condition leads to the inequality being true, ensuring all logical connections are solid and clearly explained.