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

Feb 5, 2018

Yes.

#### Explanation:

Those are linear inequalities.
Provided that the slopes of the lines are different, $e v e r y$ set of two linear inequalities has a solution that includes one sector of the intersection.

Instructions:
Sketch the graph of $y = 3 x + 5$.
As soon as you are ready to put the line onto the graph, observe that you want $y < 3 x + 5$, which is a strict inequality. Draw the line DASHED instead of solid.
Sketch the graph of $y = x + 4$.
As soon as you are ready to put the line onto the graph, observe that you want $y \ge x + 4$, which is NOT a strict inequality. Draw the line SOLID.
The ultimate solution to the system consists of the sector that you $s h a \mathrm{de} d$ both times, including part of the solid line but not the dashed line. Every point in the shaded region is a solution.