# To show:a < frac{a+b}{2} < bGiven information:a and b are real numbers.

Transformation properties

To show:
$$\displaystyle{a}\ {<}\ {\frac{{{a}+{b}}}{{{2}}}}\ {<}\ {b}$$
Given information:
a and b are real numbers.

2021-02-02

Concept used:
Number which can be written on number line called real number.
Calculation:
$$\displaystyle\text{Let}\ {a}={2}\ \text{and}\ {b}={3}$$
$$\displaystyle{a}\ {<}\ {b}$$
$$\displaystyle{2}\ {<}\ {3}$$
Then
$$\displaystyle{\frac{{{a}\ +\ {b}}}{{{2}}}}={\frac{{{2}\ +\ {3}}}{{{2}}}}$$
$$\displaystyle={\frac{{{5}}}{{{2}}}}$$
$$\displaystyle={2.5}$$
Hence, $$\displaystyle{\frac{{{a}\ +\ {b}}}{{{2}}}}$$ is in between a and b.
$$\displaystyle{a}\ {<}\ {\frac{{{a}\ +\ {b}}}{{{2}}}}\ {<}\ {b}$$
Conclusion:
For real number a and b, if $$\displaystyle{a}\ {<}\ {b}\ \text{then}{<}\ {\frac{{{a}\ +\ {b}}}{{{2}}}}\ {<}\ {b}$$