Everyone, welcome back. Up until now, we've learned how to graph individual inequalities, like this one. The basic idea was to draw the inequality, test points, and then shade the area that makes the inequality true with those points. But now we're going to look at problems that have multiple inequalities. In this case, we will repeat this process over and over, following the same steps for as many inequalities as we have. We'll see something interesting happen.

For example, if we graph an inequality, it might look like this. The shading area of this inequality will overlap with the shading area of the first graph. In this area, all the shadings will overlap, and this will be the solution to the system of inequalities. The whole idea is to graph a system of inequalities by plotting all the lines, all the curves, and shading the regions containing points that make all of the inequalities true. We'll use different colors and styles of shading, shading each curve first, and later finding the overlap. Let's follow these steps with our first example:

We've got \( y \leq -x + 4 \). First, we graph a solid line because we have a solid symbol underneath the equation. We replace it with an equal sign, and graph \( y = -x + 4 \). This line goes through the point \( (0, 4) \) with a slope of -1. To test the points, we choose any point along this area. We select the point \( (0, 2) \). Testing this point, we realize \( 2 \leq 4 \), which is a true statement. So, this area satisfies the inequality, represented as a red equation here.

We then perform the same steps for the blue equation, graphing \( y > 2x + 1 \). With a \( y \)-intercept of 1 and a slope of 2, this will be a dashed line due to the symbol. We can test the point \( (0, 2) \) again, checking if \( 2 > 1 \), which it does. Therefore, this inequality is true for this region, and we shade it accordingly.

After shading both equations, we'll find regions that overlap, showing the truly shared solution areas. We could also use different styles of hatching to distinguish between these regions until they overlap. This step will show the points that satisfy all conditions simultaneously, representing the complete solution to the system of inequalities.

One additional point: one or more equations may be non-linear. For instance, if we graph \( y \geq x^2 - 4 \), it forms a parabola with the vertex \( (0, -4) \) and shades everything above this parabola, being a solid line due to the inequality. Similarly, \( y > -x + 3 \) graphs a line through \( (0, 3) \) with a slope of -1, requiring the shading under this line. The overlap between these areas will show where both conditions fulfill.

Some systems of inequalities might have no solutions, which we'll examine later. Let me know if you have any questions regarding this process!