I decided to do some practice with some functions, and was posed with the following question:
So, I sketched the two graphs. For convenience I'll display a photo of them from Desmos.
The blue line is |$3x - 2$| and the red is |$x-5$|.
Now, to find when the two intersect, it the case where the two equations have the same output with the same input is achieved.
However, the modulus sign seemed to trip me up when I was writing |$3x - 2$| = |$x-5$| and begin doing algebra. I couldn't merely solve it like the function had no modulus sign, since that is a different function. So, I figured by inspection that only the left side of the red function makes the intersections, so the following conditions are the only valid ones.
$$-x-5 = 3x-2$$
$$-x-5 = -3x+2$$
And with that I can find the inputs necessary.
However, is there a general way to solve this type of problem? Say, if I didn't have the graph to make the inference that I did? How could I solve this problem if I couldn't graph the two functions?
You need to consider the plus-minus signs of both the equations. In the above you have considered only one. You should solve:
`$\begin{alignat*}{1} -x-5 &= 3x-2\\ -x-5 &= -(3x-2)\\ -(-x-5) &= 3x-2\\ -(-x-5) &= -(3x-2) \end{alignat*}$