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 their midpoint.

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

8

and

18

is

13,

and that the distance from both values to the midpoint is

5 .

Now we can write our equation.

∣ x - M i d p o i n t ∣ ∣ x - 13 ∣ = Distance to midpoint = 5

We can solve the equation we have created to ensure it has the desired solutions. We will need to think about both cases — the positive direction and the negative direction.

∣ x - 13 ∣ = 5

$x - 13 \geq 0 : x - 13 = 5 x - 13 < 0 : x - 13 = - 5 (I) (II)$

x - 13 = 5 x - 13 = - 5 (I) (II)

$(I), (II):$ $LHS + 13 = RHS + 13$

x_{1} = 18 x_{2} = 8

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

Exercise 31 Page 33

Hints

Check the answer

Practice exercises

Asses your skill