Problem 39 - The Calculus with Analytic Geometry

Question

Prove that if \(b>a>0\) and \(c>0\), then $$ \frac{a+c}{b+c}>\frac{a}{b} $$

Step-by-Step Solution

Answer

\( \frac{a+c}{b+c} > \frac{a}{b} \) is true if \(b > a \) and \(c > 0 \).

Step 1: Set up the inequality

Start with the given inequality: \[ \frac{a+c}{b+c} > \frac{a}{b} \]

Step 2: Cross-multiply to clear the fractions

To compare the two fractions without dealing with the fractions, cross-multiply: \[ (a+c) \times b > a \times (b+c) \]

Step 3: Expand both sides

Expand the products on both sides of the inequality: \[ ab + bc > ab + ac \]

Step 4: Simplify the inequality

Cancel the common terms on both sides. Subtract \(ab\) from both sides: \[ bc > ac \]

Step 5: Divide both sides by \(c\)

Since \(c > 0\), we can safely divide both sides by \(c\) without changing the inequality direction: \[ b > a \]

Step 6: Conclude the proof

The simplified inequality \(b > a\) is true, as it matches the initial assumption. Thus, the original inequality holds.

Key Concepts

Cross-multiplication

Cross-multiplication is a mathematical technique used to simplify equations or inequalities involving fractions. When you encounter an inequality with fractions, it can be difficult to directly compare them as is. Cross-multiplication helps by eliminating the fractions, making the comparison straightforward.

Here's a quick guide:

• Identify the two fractions you want to compare. In our case, they are \(\frac{a+c}{b+c}\) and \(\frac{a}{b}\).
• Multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction.
• The resulting inequality should no longer have fractions, making it easier to manipulate.

This technique is crucial when solving inequalities involving fractions because it simplifies the problem, allowing for more straightforward algebraic manipulation.

Inequalities

An inequality is a mathematical statement that two expressions are not necessarily equal, and one is either greater than or less than the other. In our exercise, the inequality to be proved is

\[ \frac{a+c}{b+c} > \frac{a}{b} \]

Let's break this inequality down:

• The '> sign means that the expression on the left side must be larger than the expression on the right side.
• In our context, 'a' and 'b' are constants where \(b > a > 0\), and 'c' is also a positive constant (\(c > 0\)).

To compare the two fractions, we used cross-multiplication. Understanding and solving inequalities often involve isolating variable terms on one side and ensuring the inequality remains true after each algebraic manipulation.

Algebraic Manipulation

Algebraic manipulation involves using various algebraic techniques and rules to transform and simplify expressions or equations. This step is crucial in solving inequalities like our example. After cross-multiplying, we got:

\[ (a+c) \cdot b > a \cdot (b+c) \]

Next, we expanded both sides:

• Left side: \(ab + bc\)
• Right side: \(ab + ac\)

While expanding, keep in mind the distributive property of multiplication over addition. This step allows simplifying by combining like terms:

\[ ab + bc > ab + ac \]

Finally, we simplify by subtracting common terms on both sides. In this case, subtracting \(ab\) from both sides gives us:

\[ bc > ac \]

This shows how algebraic manipulation helps isolate terms and clarify relationships between them, an essential skill for solving inequalities.

Fraction Comparison

When comparing fractions, especially under conditions given by inequalities, cross-multiplication is a handy tool. In this exercise, we needed to compare two fractions:

\[ \frac{a+c}{b+c} \: vs \: \frac{a}{b} \]

Steps to effectively compare fractions:

• Cross-multiply to clear the fractions: \( (a+c) \cdot b \) and \( a \cdot (b+c) \)
• Expand and simplify both sides of the resulting inequality.

The key here is to simplify until one side can be clearly seen to be greater than or less than the other. In our case, we demonstrated through algebraic manipulation that

\[ bc > ac \]

And since \( b > a \) by our initial assumption, this fraction comparison affirmed the truth of our original inequality. Remember, the goal is to express the problem in a way that allows you to see the relationship clearly.