Use absolute value to find the following sum.
$5 + (-9)$
The problem asks us to evaluate the sum \(5 + (-9)\) using the properties of absolute value. The absolute value of a number \(x\), written as \(|x|\), represents the distance of the number from zero on a number line, and is always non-negative. For any real number \(x\), \(|x| = x\) if \(x \ge 0\) and \(|x| = -x\) if \(x < 0\). We are dealing with two numbers: \(5\) and \(-9\). Their absolute values are \(|5| = 5\) and \(|-9| = 9\).
To add two numbers with opposite signs, we determine the sign of the result and the magnitude of the result.
First, we compare the absolute values of the two numbers: \(|5| = 5\) and \(|-9| = 9\). Since \(9 > 5\), the result of the addition will take the sign of the number with the larger absolute value. In this case, \(-9\) has the larger absolute value, so the final result will be negative.
Next, we calculate the magnitude of the sum by subtracting the smaller absolute value from the larger absolute value: \(9 - 5 = 4\).
Finally, we apply the negative sign determined in the previous step: \(-(9 - 5) = -4\). Thus, \(5 + (-9) = -4\).
Absolute value is useful in everyday scenarios involving distances and offsets. For instance, if you are playing a board game and move forward 5 squares, but then are sent back 9 squares, your net position relative to the starting square is \(-4\). The absolute value helps quantify the magnitude of the net change (4 squares back) regardless of the direction, while the sign tells you the net direction from the starting point. This is fundamental in accounting for net gain or loss in finance or navigation.