Question

For the following statement, determine whether the statement is always, sometimes or never true. Explain your reasoning.

If $a$and $b$real numbers, then $|a+b|=\left|a\right|+\left|b\right|$.

Short answer

The statement is always true.

Step-by-step solution

Step 1- Consider the statement.

Consider the statement $|a+b|=\left|a\right|+\left|b\right|$.

Step 2- Step description.

Let $a=5,b=3$.

Substitute the values $a=5,b=3$in the equation $|a+b|=\left|a\right|+\left|b\right|$and simplify.

$\begin{array}{l}|a+b|=\left|a\right|+\left|b\right|\\ |5+3|=\left|5\right|+\left|3\right|\\ \left|8\right|=5+3\\ 8=8\end{array}$

Step 3- Step description.

Since the values are same on both the sides (L.H.S=R.H.S.) thus, the statement is always true.

Therefore, the statement is always true.