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-Midpoint|&= Distance to midpoint
|x- 2|&= 8
We can solve the equation we have created to ensure it has the desired solutions.
|x-2|=8
lc x-2 ≥ 0:x-2 = 8 & (I) x-2 < 0:x-2 = - 8 & (II)
lcx-2=8 & (I) x-2=- 8 & (II)
(I), (II): LHS+2=RHS+2
lx_1=10 x_2=- 6
Keep in mind that this is just one possible absolute value equation we could use.
