How do I read modulus graphs?

[Kulla_9289]

How do I read modulus graphs? I have two functions drawn on a graph, and using that I need to solve |x-2|>|2x-10|. I am aware of solving this using an algebraic method by squaring both sides or separating them into two inequalities.

 

[Harry_the_cat]

The blue graph is y=abs(x-2).
The red graph is y=abs(2x-10).

If you want to find , graphically, where abs(x-2) = abs(2x-10), you would look for the x-coordinates of the points of intersection of the blue graph and the red graph(ie x=4 or 8).

If you want to find when abs(x-2) > abs(2x-10), you need to look at where the blue graph lies above (ie >) the red graph.
Can you see, for what x-values, the blue graph lies above the red graph? It will be an interval of points, such as ?? < x < ??

Note: This is the same as what you would do for normal graphs without involving absolute values when dealing with inequalities rather than equalities (equations).

 

[Kulla_9289]

I get this. But what if I had [imath]\left|2x+1\right|\ge\left|3-x\right|[/imath]? Why is it not in an interval of points?

 

[Harry_the_cat]

Red graph is y=abs(2x+1).
Blue graph is y = abs(3-x).

For abs(2x+1) >=abs(3-x), we need the red curve above or on the blue curve.
This happens when x<=-4 and when x>=2/3.

It is a set of values, just not a continuous interval.

 

[Kulla_9289]

Yep. I get this now. Thank you.