Find the graph of the inequality y< 3/4x+2

The first thing that we need to do when solving a linear inequality, is drawing the line:

That is, we must draw the line y=(3/4)x+2.

We know that two points are enough: For example, for x=0, we immediately get y=2.

While, for y=0, we have (3/4)x=-2, that is 3x=-8, which yields x=-8/3.

Thus, we got (0, 2) and (-8/3, 0), which are the x-intercept and the y-intercept, respectively.

We draw the line through these lines, but it must be dashed since the points on this line are solutions of the equation y=(3/4)x+0, NOT solutions of the inequality.

Now, we can check which side to color, by picking a point not on the line, for example (0, 5), as follows:
 
for x=0, y=5, substituting into the inequality we have 5<2, which is not true. So, the part of the plane above the line is not a solution, but rather, the lower part is.

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