Question

Solve the following equation 2|x-3|+5=17

Answer 1

To solve this equation, we can start by solve for the absolute value part, the |x - 3|:

`\begin{gathered} 2|x-3|+5=17 \\ 2|x-3|=17-5 \\ 2|x-3|=12 \\ |x-3|=(12)/(2) \\ |x-3|=6 \end{gathered}`

Now, we have to consider two possibilities:

`\begin{gathered} x-3\ge0 \\ x-3<0 \end{gathered}`

If the first one is true, then the absolute value function won't change the part inside it, so we can just remove the vertical bars.

If the second one is true, then the absolute value function will change the sign of the part that is inside it, so we can change the sign of the part inside and then remove the vertical bars.

So, we have:

`\begin{gathered} if\, x-3\ge0\colon \\ |x-3|=x-3 \\ x-3=6 \\ x=6+3 \\ x=9 \end{gathered}`

And:

`\begin{gathered} if\, x-<0\colon \\ |x-3|=-(x-3) \\ -(x-3)=6 \\ -x+3=6 \\ -x=6-3 \\ -x=3 \\ x=-3 \end{gathered}`

So, the solutions for the given equation are:

`\begin{gathered} x=9 \\ x=-3 \end{gathered}`