Question

Arithmetic Mean If \(a

Short answer

\(a < \frac{a+b}{2} < b\)

Step-by-step solution

- State the Given Inequality

Start with the given assumption: \(a < b\)

- Add 'a' to Both Sides

To move towards forming an average, add 'a' to both sides of the inequality: \(a + a < a + b\)This simplifies to: \(2a < a + b\)

- Divide by 2

Now, divide every term in the inequality by 2 to obtain: \(\frac{2a}{2} < \frac{a+b}{2}\)Which simplifies to: \(a < \frac{a+b}{2}\)

- Add 'b' to Both Sides

Next, to establish the upper bound, add 'b' to the original assumption: \(a < b\)Add 'b' to both sides: \(a + b < b + b\)This simplifies to: \(a + b < 2b\)

- Divide by 2 Again

Now divide every term in the new inequality by 2: \(\frac{a+b}{2} < \frac{2b}{2}\)Which simplifies to: \( \frac{a+b}{2} < b\)

- Combine Results

Combining the results from the previous steps, we obtain: \(a < \frac{a+b}{2} < b\)

Key concepts

Inequality Proof

To prove the inequality \(a < \frac{a+b}{2} < b\), it's important to understand what an inequality proof entails. Here, we are given that \(a < b\), and we need to show that the arithmetic mean \( \frac{a + b}{2} \) falls between \(a\) and \(b\). By performing a series of algebraic manipulations on the given inequality, we can confirm that it holds true. The key steps generally involve adding or subtracting the same quantity on both sides of the inequality, or multiplying/dividing by positive numbers. This process maintains the direction of the inequalities, ensuring accurate results.

The main goal of an inequality proof is to use logical reasoning to establish the required inequalities rigorously. Understanding each manipulation's impact can make the proof more intuitive and straightforward.

Mean Calculation

The arithmetic mean, or average, of two numbers \(a\) and \(b\) is given by \(\frac{a + b}{2}\). This mean represents a value that balances these two numbers, and it is always between them if \(a < b\). In this exercise, this mean calculation helps to affirm the central value between \(a\) and \(b\).

Calculating an arithmetic mean is straightforward:

• Add the two values: \(a + b\)
  
• Divide the sum by 2: \(\frac{a + b}{2}\)
  

For example, if \(a = 3\) and \(b = 7\), the arithmetic mean is \(\frac{3+7}{2} = 5\). This result lies between 3 and 7, illustrating our initial point. Understanding the mean calculation helps make the inequality more intuitive, showing how averages work in elementary mathematics.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations or inequalities to form a desired expression. It's essential for solving or proving mathematical statements.

In this exercise, we use several manipulations:

1. Adding the same number to both sides of an inequality: From \(\text{if } a < b, \text{ then } a + a < a + b\). This preserves the inequality direction.
   
2. Dividing both sides by a positive number: From \(2a < a + b \text{ to } a < \frac{a+b}{2}\).
   

Remembering these rules for algebraic manipulation can make solving inequalities more approachable and less intimidating. Manipulating inequalities is fundamental to many mathematical proofs and problem-solving techniques.

Mathematical Inequalities

Mathematical inequalities express the relative size or order of two values. Symbols like \( <\) and \( > \) allow us to compare numbers and establish relationships between them. Understanding how these symbols behave under various operations is key.

A few critical properties of inequalities include:

• Adding or subtracting the same value on both sides doesn't change the inequality's direction.
  
• Multiplying or dividing both sides by a positive number maintains the inequality. Dividing by a negative number flips the inequality.
  

These properties help us manipulate inequalities correctly, as we did in the exercise to establish that \(a < \frac{a + b}{2} < b\). Such concepts are instrumental in various branches of mathematics, including calculus and linear programming.