Chapter 2.7, Problem 17WE

To determine

To prove:

If $a<b$ then $a<\frac{a+b}{2}<b$

Explanation of Solution

Given information:

  $a<b$

Proof:

Here, the given proof statement is:

If $a<b$ then $a<\frac{a+b}{2}<b$

1. Given that:

  $a<b$............ (1)

2. Use the addition property. Add $a$ on both sides of inequality equation as:

  $a+a<b+a$

3. Use the distributive property to collect the like terms together as:

  $2a<b+a$

4. Use commutative property of addition as:

  $2a<a+b$

5. Use the first multiplication property. Multiply by $\frac{1}{2}$ on both sides of equation

  $\frac{1}{2}\cdot \left(2a\right)<\frac{1}{2}\cdot \left(a+b\right)$

6. Now, use property of reciprocals and the definition of division to get the result as:

  $a<\frac{a+b}{2}$

7. Use the addition property. Add $b$ on both sides of inequality equation (1):

  $a+b<b+b$

8. Use the distributive property to collect the like terms together as:

  $a+b<2b$

9. Use the first multiplication property. Multiply by $\frac{1}{2}$ on both sides of above equation:

  $\frac{1}{2}\cdot \left(a+b\right)<\frac{1}{2}\cdot 2b$

10. Now, use property of reciprocals and the definition of division to get the result as:

  $\frac{a+b}{2}<b$

11. As per the transitive property:

  $a<\frac{a+b}{2}<b$

Hence proved