In this exercise, we have been given a system consisting of two linear inequalities. We'll tackle them one at a time and bring them together in a system at the end. Let's call the inequalities and
It only takes two points to create a unique equation for any line, so let's start by identifying two points on the boundary line.
Here we've identified two points, and and indicated the horizontal and vertical changes between them. This gives us the rise and run of the graph. We can use this information to find the slope, One of the points we selected, is also the -intercept. With slope and the -intercept a we can write an equation for the boundary line in slope-intercept form. Now that we found the boundary line we should find the correct inequality symbol. To figure out what the symbol should be, let's substitute any point that lies within the solution set into the equation.
Writing the inequality for this region of the graph will involve the same steps as above. We'll start by identifying two points.
Again, we have denoted the rise and run of the graph, giving the slope Since we chose the -intercept at the point as one of our points for this boundary line as well, we can write its equation. Once more, we replace the equality sign with a question mark. To determine the correct inequaality symbol, we will use a test point.
First we need to identify two points on the line where one preferably is the -intercept.
Here we've identified two points, and and indicated the horizontal and vertical changes between them. This gives us the rise and run of the graph, which will give us the slope, Now that we have the slope and the -intercept we can write an equation of the line in slope-intercept form. We should now determine the correct inequality symbol. This is done using a test point. Let's use the origin.
The boundary line of the second region of the graph is a vertical line. Therefore, writing an inequality for this region requires a different approach.
Notice that every point on the boundary line will have a -cordinate equal to This information is enough to write corresponding equation. Since the region to the left of the line is shaded it represents all -values less than We can now write our system of inequalities.