Question

What is an absolute value equation that has the solutions $x=8$ and $x=18?$

Answer 1

Hint: We know that an absolute value equation is an equation in which the variable is enclosed in the absolute value operator. And we also know that the absolute value operator produces only positive values.

Complete step-by-step answer:
Let us consider the given problem.
We are asked to find an absolute value equation for which $x=8$ and $x=18$ are two solutions.
So, in this case, we need to find out an equation in which the variable $x$ is enclosed within the absolute value operator modulus which provides the value with positive sign regardless of the sign of the value before the operator is applied.
So, here, we need to find an absolute value equation which satisfied by $x=8$ and $x=18.$
We need to find a number that is common to these two solutions in the sense that the number is the same distance from both $8$ and $18.$
Let us find the average of the above two numbers.
Then, we will get $\dfrac{8+18}{2}=\dfrac{26}{2}=13.$
So, we know that $13$ is a number that is the same distance from both the numbers $8$ and $18.$
We know that when $x=8,$ then will get $x-13=8-13=-5.$
Also, when $x=18,$ then we will get $x-13=18-13=5.$
We can say that $\left| x-13 \right|=5.$
Now, when we apply $x=8,$ we will get $\left| 8-13 \right|=\left| -5 \right|=5.$
When we apply $x=18,$ we will get $\left| 18-13 \right|=\left| 5 \right|=5.$
And now we will get an absolute value equation $\left| x-13 \right|=5.$
Hence the absolute value equation for which $x=8$ and $x=18$ is $\left| x-13 \right|=5.$

Note: We know that the absolute value of a number $x$ can be defined as a function $f$ from the set of real numbers $\mathbb{R}$ to itself $f:\mathbb{R}\to \mathbb{R}$ by $f\left( x \right)=\left\{ \begin{align}
  & x,\,\,\,\,if\,x\ge 0 \\
 & -x,\,\,if\,x<0 \\
\end{align} \right.$ and we denote the absolute value function as $\left| x \right|.$