Challenge: Find values of a and b for which the statement is true.

|a + b| = |a| + |b|

Absolute Value

Absolute value signs make the final result of the operations between them positive. When deciding on the order of operations, treat absolute value bars like parentheses.

|– 3| = 3: The absolute value bars make negative 3 positive.

| 3 | = 3: The absolute value bars leave positive 3 unchanged

– |– 3| = – 3: The absolute value bars make negative 3 positive. Then the negative sign takes effect, and the answer is a negative 3.

– | 3 | = – 3: The absolute value bars leave positive 3 unchanged. Then the negative sign takes effect, and the answer is a negative 3.

The simplest case being when both a and b are non negative; in this case$|a+b|=a+b=\left|a\right|+\left|b\right|$

To find the values of a and b, four cases are considered.

Case 1: Both a and b are positive.$\begin{array}{l}|a+b|=\left|a\right|+\left|b\right|\\ a+b=a+b\end{array}$

Example: a = 2 and b = 3. $\begin{array}{l}|a+b|=a+b=\left|a\right|+\left|b\right|\\ |a+b|=\left|a\right|+\left|b\right|\\ 2+3=2+3\\ 5=5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}True}\end{array}$

Case 2: Both a and b are negative$\begin{array}{l}\left|(-a)+(-b)\right|=|-a|+|-b|\\ |-a-b|=|-a|+|-b|\\ \left|-(a+b)\right|=|-a|+|-b|\\ a+b=a+b\end{array}$

Example: a = – 2 and b = – 3$\begin{array}{l}|a+b|=\left|a\right|+\left|b\right|\\ \left|(-2)+(-3)\right|=\left|(-2)\right|+\left|(-3)\right|\\ |-5|=|-2|+|-3|\\ 5=2+3\\ 5=5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}True}\end{array}$

Case 3:a is positive,b is negative.$\begin{array}{l}\left|a+(-b)\right|=\left|a\right|+|-b|\\ |a-b|=\left|a\right|+|-b|\\ a-b\ne a+b\end{array}$

Example: a = 2 and b = – 3$\begin{array}{l}|a+b|=\left|a\right|+\left|b\right|\\ \left|2+(-3)\right|=\left|2\right|+\left|(-3)\right|\\ |-1|=\left|2\right|+|-3|\\ -1=2+3\\ -1\ne 5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}False}\end{array}$

Case 4:ais negative and b is positive.$\begin{array}{l}\left|(-a)+b)\right|=|-a|+|-b|\\ |-a+b|=|-a|+\left|b\right|\\ b-a\ne b+a\end{array}$

Example: a = – 2 and b = 3$\begin{array}{l}|a+b|=\left|a\right|+\left|b\right|\\ |-2+3|=|-2|+\left|3\right|\\ \left|1\right|=|-2|+\left|3\right|\\ 1=2+3\\ 1\ne 5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}False}\end{array}$

Since the equation is satisfied in cases 1 and 2, it can be concluded that the values of a and b will have the same sign