Question

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

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

Short answer

The statement is always true.

Step-by-step solution

Step 1- Consider the statement.

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

Step 2- Step description.

Let $a=5,b=3,c=2$.

Substitute the values $a=5,b=3,c=2$in the equation $c|a+b|=\left|ca\right|+\left|cb\right|$and simplify.

$\begin{array}{l}c|a+b|=\left|ca\right|+\left|cb\right|\\ 2|5+3|=\left|\left(2\right)5\right|+\left|\left(2\right)3\right|\\ 2\left|8\right|=\left|10\right|+\left|6\right|\\ 2\left(8\right)=10+6\\ 16=16\end{array}$

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.