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 at most 4. The y-variable added to the product of 3 and the x-variable does not exceed 6.

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 B solution set here, our given statements are Y added to X yields a sum that is at most seven. The product of four and XX added to Y cannot be greater than 11. Here we have four answer choice options. First answer choice A which has two solid lines joined at some 20.1 line crosses through the 0.0 11. And the area above both of the lines is shaded. Next, we have answer choice B which also has two solid lines joined at some 20.1 line crosses through the .07. And the area beneath the two lines is shaded. Then we have answer choice C which is another graph but only with one line. This solid line passes through the .011 and there is another point displayed at 07 and the area underneath the line and between the two points is shaded. And our last answer choice is choice D no solution. So moving back to our given statements, we want to begin this problem by first writing inequalities that describe each of our statements. So beginning with the first statement, which is that Y added to X yields a sum that is at most seven, we can break up this statement and write it as an inequality. So first, we have Y added to X, which means we have Y plus X and this yields a sum that is at most and at most tells us that this expression is less than or equal to. And then the value given is seven. So we have Y plus X is less than or equal to seven. Now moving on to our second statement, we have the product of four and X. So we, the product of four and X tells us we have four X added to Y. So we add Y four, X plus Y cannot be greater than 11 while they cannot be greater then tells us that this expression for X plus Y is less than or equal to the value given 11. So now we have written our system of inequalities, our 22 given inequalities for each statement. And now we can move on to graphing these solution set. So to graph these solutions set, we want to first graph each inequality individually. So first, again, if we recall, we have Y plus X is less than or equal to seven. So now graphing this inequality, we get the graph of a line that is decreasing. It is a solid line due to the equal sign and the inequality and it passes through the .011. And now because our inequality is the less than symbol or has the less than symbol, the area beneath the line is shaded. And now that we have graphed, our first inequality, we also need to graph the second inequality which if we recall is four X plus Y is less than or equal to 11. Now graphing our inequality, we get the graph of another decreasing line. This line is also solid due to the equal sign and the inequality it passes through the .07. And because the expression is less than 11, we have the area beneath the line shaded. And now that we have graphed both all of our inequalities individually, we just need to combine both of the graphs to get our graph of these solutions set. So with this new graph, combining both of our graphs, we see we now have two lines joined at some point, both lines are solid and the, the top line passes through the .07. And then the area beneath both of the lines just like they are in their individual graphs are shaded in. So our solution with this graph is the shaded region along with both of the lines since they are solid. So with this graph, we can compare it to our possible answers here. And we see that we are left with answer choice B 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 at most 4. The y-variable added to the product of 3 and the x-variable does not exceed 6.

Key Concepts

Inequalities

Systems of Inequalities

Graphing Inequalities

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