The graph of |2x - 4| is upward sloping and has a gradient of 2 for x > 2. For this portion of the graph, y = 2x - 4

It is downward sloping and has a gradient of -2 for x < 2. For this part, y = 4 - 2x

If m = 2 , the line y = mx - 1 would be y = 2x - 1, and would be parallel to the line where y = 2x - 4. They will not intersect. y = 2x - 1 will only intersect the portion y = 4 - 2x for this gradient. So that's only one point of intersection.


So the line y = mx - 1 has to be less steep in order to cut at 2 intersection points. This means m < 2


But if the line becomes less steep, it will eventually intersect y = |2x - 4| at only 1 point again and this is at the point (2,0), the point where the graph changes from y = 4 - 2x to y = 2x - 4

sub (2,0) into y = mx - 1,

0 = m(2) - 1
2m = 1
m = ½

So the value of m here is ½ and there is only one point of intersection. If it is even less steep, there will be 0 points of intersection.

So the line must be steeper than that, meaning m > ½


Put the two together,

½ < m < 2