X-Intercepts of Absolute Value Function y = 2 |x - 3| + 1

[JohLo]

Hello, I'm having what I assume is a simple problem of misunderstanding something basic. I want to figure out the x-intercepts of the equation y=2|x-3|+1. I know what the graph looks like. I know it doesn't have any x-intercepts. But when I try calculate the x-intercepts, I'm getting answers that tell me it should have x-intercepts and I don't know what I'm doing wrong.

My process:
y=2|x-3|+1
Substitute 0 for y.
0=2|x-3|+1
-(1/2)=|x-3|
-(1/2)=x-3 OR -(-(1/2)=x-3
5/2=x OR 7/2=x

The math I see says I should have x-ints at (5/2, 0) and (7/2, 0). But that can't be right because the V of the graph is shifted up by 1 unit relative to the x axis.

I thank you kindly for helping me understand this paradox.

 

[Steven G]

Have you checked your possible answers to see if any of them worked.

You should know that the absolute value of anything is never negative. So when you had -(1/2)=|x-3| you should have stopped right there! If you insist on proceeding from there then you MUST check your solutions as they WILL (both) fail!

 

[pka]

Hello, I'm having what I assume is a simple problem of misunderstanding something basic. I want to figure out the x-intercepts of the equation \(\displaystyle y=2|x-3|+1\). I know what the graph looks like. I know it doesn't have any x-intercepts. But when I try calculate the x-intercepts, I'm getting answers that tell me it should have x-intercepts and I don't know what I'm doing wrong.

Do you agree that \(\displaystyle 2|x-3|\ge 0~?\)
Now if you do, then you would agree that \(\displaystyle 2|x-3|+1\ge 1\). What does that imply about any \(\displaystyle x-\text{intercept}~?\)

 

[Steven G]

Just out of curiosity why did you even try to solve this equation if you knew there was no solution?

 

[JohLo]

Argh, yes of course, I knew it was something silly! Thank you for your help!

Jomo, I am concerned with understanding all the ins and outs of the method, not just getting the answer correct. If I were asked a question about that method, I could only just sit there, mute. In addition, what if I didn't have the luxury of foreknowledge? Then I would only have the method to lead me out from the wilderness of my ignorance. Your and pka's aid reminds me to lift my head up out of computation and think about what I'm doing and what it means, rather than just crunching numbers like a calculator, receiving "ERROR", and giving up.

Again, much obliged.