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

- 10

and

- 5

is

- 7.5,

and that the distance from both values to the midpoint is

2.5 .

Now we can rewrite our equation.

∣ x - M i d p o i n t ∣ ∣ x - (- 7.5) ∣ = Distance to midpoint = 2.5 \Leftrightarrow ∣ x + 7.5 ∣ = 2.5

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

∣ x + 7.5 ∣ = 2.5

$x + 7.5 \geq 0 : x + 7.5 = 2.5 x + 7.5 < 0 : x + 7.5 = - 2.5 (I) (II)$

x + 7.5 = 2.5 x + 7.5 = - 2.5 (I) (II)

$(I), (II):$ $LHS - 7.5 = RHS - 7.5$

x_{1} = - 5 x_{2} = - 10

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

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