How do you solve graphically #abs(x – 4)>abs(3x – 1)#?

graph{(y-abs(x-4))(y-abs(3x-1))=0 [-20.28, 20.27, -10.14, 10.13]}

Therefore, we need to find the point of intersection

To solve graphically the inequality abs(x – 4) > abs(3x – 1), you would plot the graphs of the two expressions abs(x – 4) and abs(3x – 1) on the same coordinate plane. Then, identify the regions where the graph of abs(x – 4) is greater than the graph of abs(3x – 1). These regions represent the solutions to the inequality.

To solve the inequality ( |x - 4| > |3x - 1| ) graphically, follow these steps:

1. Graph the functions ( y = |x - 4| ) and ( y = |3x - 1| ) on the same coordinate system.
2. Identify the regions where one function's value is greater than the other function's value.
3. The solution to the inequality lies in the regions where ( |x - 4| ) is greater than ( |3x - 1| ).

Here's a step-by-step process:

1. Graph ( y = |x - 4| ):

   • This function represents the absolute value of ( x - 4 ), which is symmetric around the point ( x = 4 ). It is a V-shaped graph with the vertex at ( (4, 0) ).

2. Graph ( y = |3x - 1| ):

   • This function represents the absolute value of ( 3x - 1 ), which is symmetric around the point ( x = \frac{1}{3} ). It is also a V-shaped graph, but steeper than the first one, with the vertex at ( \left(\frac{1}{3}, 0\right)).

3. Identify the regions where ( |x - 4| ) is greater than ( |3x - 1| ):

   • Look for the regions where the graph of ( |x - 4| ) is above the graph of ( |3x - 1| ). This typically involves identifying the areas to the left and right of the intersection point(s) of the two graphs.

4. Determine the solution:

   • The solution to the inequality is the set of ( x ) values corresponding to the identified regions.

Remember that in a graphical solution, you're looking for the regions where the graph of the left side of the inequality is higher (or lower, depending on the inequality) than the graph of the right side.