In Exercises 27–62, graph the solution set of each system of ineq...

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x−y≤2, x>−2, y≤3

Math inequalities for graphing: x - y ≤ -3, x > -5, y ≤ 1.

Accepted Answer

Hello, today we're going to solve the system of inequalities by using the method of graphing. So we are given a system of inequalities of three inequalities. I'm gonna go ahead and label each of the inequalities and we're gonna take a look at each of their graphs individually. So if we're looking at the first inequality, we are given X minus Y is less than or equal to negative three. Now we can use the method of graphing lines to rewrite this to match the form of the equation of a line. See, we can imagine this as a standard form of the equation of the line X minus Y is equal to negative three. In this case here, if we wanted to rewrite this into the slope intercept form, we would want to solve for our Y variable. So we're gonna go ahead and use that same method and solve for our Y variable in inequality. So the first thing we can do is subtract X to both sides of the inequality that is going to give us negative Y is less than or equal to negative X minus three. And to solve for Y, we're going to divide both sides of the equation by negative one. But one thing to note is that when you divide an inequality by negative one, you must flip the direction of the sign. So dividing by negative one is going to give us the inequality Y is greater than or equal to X plus three. So this matches the form of a line where it starts at three on the Y axis and has a slope of positive one. So if we draw an axis, the starting point of this line is going to be a positive three, no, the question is do we want to graph this using a solid or dotted line? While this inequality is saying that we want all the values that are greater than or equal to Y. So that means we're going to include the values on the number line. Therefore, we're gonna graph this positive line using a solid line. And one more question is do we want to include the region underneath as our solution for this inequality or the region above the line as a solution? Well, the easiest way to determine this is to pick a testing point from above or below the line. If we plug it into our inequality and it becomes a true statement, then we're going to shade that side of the line. The easiest testing point to pick in this case is going to be the origin zero comma zero. And if we plug this into our inequality, we get zero is greater than equal to zero plus three. Simplifying the right hand side is going to give us zero is greater than or equal to three, but this is a false inequality. So since the origin is located below the line, that means we want to shade the region above the line since the origin is not considered a valid solution. So that's going to be the graph for the equation of the first inequality. Now let's take a look at the second inequality. The second inequality is given to us as X is greater than negative five. Now this matches the form of the equation of a vertical line. X is equal to negative five. This is the equation of a vertical line where it starts at negative five on the X axis. But again, do we want to represent this using a solid or dotted line? Well, this is using the inequality symbol greater then and since it's greater than and not greater than or equal to, we're gonna represent this vertical line using a dotted line. And one more thing to note is that we want all the X values that are greater than negative five. That means we want all the values greater than negative five, which is going to be the right hand region from the line. So we're going to shade in the right region of this dotted line. And that's going to be the graph for this inequality. Next, we're going to take a look at the third inequality. The third inequality is given to us as Y less than or equal to one. Now, again, this matches the equation of a horizontal line Y is equal to one. So if we were to plot this on a graph, we can represent this as a horizontal line starting at positive one. But again, do we want a dotted or solid line? While this inequality is stating that we want less than or equal to one. So we want to include the values that lie on the horizontal line. Therefore, we're going to plot this using a solid line. And what the inequality statement is saying is that we want all the values of Y that are less than or equal to one. So all the values that are less than are equal to one on the Y axis are going to be below the solid line. So we're going to want to shade the solid line underneath or shade the region underneath the solid line. Now, if we combine all these graphs into a single plane, we can go ahead and get our solutions. So the third inequality had a solid line, a solid horizontal line starting at one on the Y axis. The second inequality had a dotted line, a dotted vertical line at negative five. And we had a positive slope line which was solid starting at three and cutting across. And if we were to just focus on the overlapping shaded region. We're going to end up getting a triangular shape for our solution. So now we just need to select the option that accurately represents this information. And that option is going to be a, just to verify we have a solid horizontal line that starts at one on the ax on the y axis, we have a dotted vertical line which starts at negative five on the X axis and we have a solid line which starts at three on the y axis. So with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x−y≤2, x>−2, y≤3

Key Concepts

Inequalities

Graphing Linear Inequalities

Systems of Inequalities

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