Question

I need to find an equation using absolute value that will give me the answer of both 8 and 14. Please help!

Answer 1

We have to write an equation with an absolute value that will give the same answer for x=8 and x=14.

We can write a generic equation with a parameter and then find its value:

`\begin{gathered} f(x)=|x-a| \\ f(8)=f(14) \end{gathered}`

Then:

`\begin{gathered} f(8)=|8-a| \\ f(14)=|14-a| \\ \Rightarrow|8-a|=|14-a| \end{gathered}`

One of the two arguments, "8-a" or "14-a", has to be negative and the other positive. It must be "8-a" because if "14-a" were negative, then "8-a" would be also negative and there is no value of a that can make the two terms equal.

Then, we can rewrite the equality as:

`-(8-a)=14-a`

we then can solve for "a" as:

`\begin{gathered} -(8-a)=14-a \\ a-8=14-a \\ a+a=14+8 \\ 2a=22 \\ a=(22)/(2) \\ a=11 \end{gathered}`

Note that x=11 is the midpoint between x=8 and x=14.

Answer:

The absolute value function that gives the same value for x=8 and x=14 is f(x) = |x-11|