In Exercises 65–68, write the given sentences as a system of ineq...

Question

In Exercises 65–68, write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the x-variable and the y-variable is no more than 2. The y-variabe is no less than the difference between the square of the x-variable and 4.

Accepted Answer

Hey, everyone. Here we are asked to express the following statements as a system of inequalities in two variables and using one Cartesian coordinate system graph the solution set. So here our given statements are Y added to X yields a sum that is no more than five and Y is no less than the square of X decreased by nine. So here we have four answer choice options. First answer choice A which is a graph with a Parabola which is a solid line as well as a line that is also solid and passes through the 40.5. And then the Parabola passes through the points negative 300, negative nine and 30. And then the area enclosed by the line and the Parabola is shaded. Next with answer choice B, we just have a line passing through the .05 and the area above the line is shaded. Next we have answer choice C which is just the upward facing Parabola passing through the points negative 300, negative nine and 30. And then the area underneath the solid line. Parabola is shaded. And lastly answer choice D where we simply have no solution. So now looking back at our given statements, we want to begin this problem by first writing a system of inequalities for these given statements. So starting with the first statement where again, we have Y added to X yields a sum that is no more than five, five. So breaking up this statement, we have first Y added to X, which tells us we have Y plus X and then it yields a sum that is no more than so it must be less than, or equal to five. So we have again, Y plus X is less than, or equal to five. Now, looking at our second given statement, we have Y is no less than the square of X decreased by nine. Breaking it up, we have first Y is no less than, which means Y must be greater than or equal, equal to the square of X is X squared decreased by nine, tells us X squared decreases by nine. So we have X squared minus nine. So our second inequality is Y is greater than, or equal to X squared minus nine. So now that we have identified our system of inequalities, we just want to move on by graphing each individual inequality. So beginning with our first inequality which again was Y plus X is less than or equal to five. And graphing this inequality, we see that we have a decreasing solid line that passes through the points +05. And +50. And then because of the less than sign, we have the area underneath the line shaded in. And now that we have graphed our first inequality, we can move on to just graphing the second inequality which was Y is greater than or equal to X squared minus nine. So graphing we end up with an upward facing parabola passing through the points negative three 00, negative nine and 30. And because of our greater than sign in the inequality, we have the shaded area, the the area above the Parabola shaded in and because of the equal sign and the inequality, the line of the Parabola is solid. So now that we have graphed both of our inequalities, we can simply combine our graphs to come up with our solution graph. So combining our two graphs and including the boundaries from both of the graphs, we have the upward facing Parabola still passing through the points negative 300, negative nine and 30. And the Parabola is still a solid line. And then we have the line from our first graph which now only passes through the points 05 and it is still a solid line as well. And then the area enclosed by the line and the Parabola is shaded. So here combining the graphs, we see our solution are both the solid line Parabola and the solid line as well as the area shaded in between the two lines So with our combined graph, we see we are left here with answer choice a where again, we have a parabola and a line and the area enclosed between them is shaded in. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 65–68, write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the x-variable and the y-variable is no more than 2. The y-variabe is no less than the difference between the square of the x-variable and 4.

Key Concepts

Inequalities

Graphing Systems of Inequalities

Quadratic Functions

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