Exercise 32 Page 33

To write an absolute value equation, we can begin by thinking about the solutions to the equation as points on a number line. We can use the number line to determine the midpoint and the distance from each point to the midpoint. Our equation will take the following form.

∣ x - Midpoint ∣ = Distance to midpoint

Let's plot the given solutions on a number line and determine the midpoint.

From the number line, we can see that the midpoint between

- 6

and

10

is

2,

and that the distance from both values to the midpoint is

8 .

Let's write our equation.

∣ x - M i d p o i n t ∣ ∣ x - 2 ∣ = Distance to midpoint = 8

We can solve the equation we have created to ensure it has the desired solutions.

∣ x - 2 ∣ = 8

$x - 2 \geq 0 : x - 2 = 8 x - 2 < 0 : x - 2 = - 8 (I) (II)$

x - 2 = 8 x - 2 = - 8 (I) (II)

$(I), (II):$ $LHS + 2 = RHS + 2$

x_{1} = 10 x_{2} = - 6

Keep in mind that this is just one possible absolute value equation we could use.