Question

Are there solutions to the system of inequalities described by #y<3x+5, y>=x+4#?

Answer 1

Yes.

Explanation:

Those are linear inequalities.
Provided that the slopes of the lines are different, `every` set of two linear inequalities has a solution that includes one sector of the intersection.

Instructions:
Sketch the graph of `y = 3x + 5`.
As soon as you are ready to put the line onto the graph, observe that you want `y < 3x + 5`, which is a strict inequality. Draw the line DASHED instead of solid.
Shade lightly below the line.

Sketch the graph of `y = x + 4`.
As soon as you are ready to put the line onto the graph, observe that you want `y >= x + 4`, which is NOT a strict inequality. Draw the line SOLID.
Shade lightly above the line.

The ultimate solution to the system consists of the sector that you `shaded` both times, including part of the solid line but not the dashed line. Every point in the shaded region is a solution.

[For example, notice that (1, 6) is a solution to both inequalities.]