Exercise 51 Page 107

We want to write an absolute value equation for the given graph.

Let's first determine the values of the given points, and then write an equation including absolute value for this graph.

Finding the Values of the Points

Note that the segments between each whole number on the number line are divided into $3$ equal spaces. It shows the denominator of any point on the number line.

In order to find the numerator of our numbers, we need to look at how many steps we move away from $0.$

Therefore, one of our numbers is a fraction with a numerator of $2$ and a denominator of $3,$ that is, $\frac{2}{3}.$ The other number is also a fraction, but since we move to the negative direction to reach it, it is a negative fraction with a numerator $4$ and a denominator $3,$ that is, $\text{-}\frac{4}{3}.$

Writing an Absolute Value Equation

Absolute values can be interpreted as the distance away from a midpoint. For one-variable absolute value equations, this distance can be represented by two points on a number line, such as the ones given in the exercise.

Because our equation needs a $\text{distance}$ from a $\text{midpoint},$ we should begin by finding the halfway point between the two given values. We can do this by calculating their mean. Our midpoint is $\text{-}\frac{1}{3}.$ Now, we need to find the distance between this midpoint and each of the given points. We see that both $\text{-}\frac{4}{3}$ and $\frac{2}{3}$ are $1$ unit away from $\text{-}\frac{1}{3}.$ Written as an equation, we can show that the difference between a number $x$ and the midpoint is equal to the distance we found above.