In Exercises 69–70, rewrite each inequality in the system with...

Question

In Exercises 69–70, rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. |x|≤2, |y|≤3

Inequalities |x| ≤ 5 and |y| ≤ 7 displayed in a mathematical format.

Accepted Answer

Hey, everyone here we are asked to write the following system of inequalities into equivalent expressions but without the absolute value bars and then graph the resulting system in the same Cartesian plane. So here our given system of inequalities are the absolute value of X is less than or equal to five and then the absolute value of Y is less than or equal to seven. Here we have four answer cho choice options. First answer choice A which displays a graph of the system of inequalities negative five is less than or equal to X which is less than or equal to five. Then negative seven is less than, or equal to Y which is less than, or equal to seven. Then we have answer choice B which is a graph of the system of inequalities X is less than or equal to five and Y is less than or equal to seven. Then we have answer choice C which is a graph of the system of inequalities negative five is less than X which is less than five. Then negative seven is less than Y which is less than seven. And lastly, we have answer choice D which says we have no solution. So now moving back up to our given system of inequalities with the absolute value sign. Again, first, we need to rewrite this system without the absolute values. So we can begin with our first inequality where we have the absolute value of X is less than or equal to five. So this tells us that we can take negative values and positive values of X as long as the result is less than or equal to five. So that means that we can go from negative 5 to 5 as values for X. So we can have negative five is less than or equal to X which is less than or equal to five. And now repeating this for the second inequality where we have the absolute value of Y is less than or equal to seven. So here we see that the minimum value of Y that we can input into this in inequality is negative seven. So we can have negative seven is less than or equal to Y which is less than or equal to seven. So now that we have rewritten our system of inequalities without the absolute value bars, we can move on to finding a graph that graphs both of these inequalities. So we can by first just graphing the first inequality which is again negative five is less than or equal to X which is less than or equal to five. So now graphing this inequality we have this vertical rectangle image on the graph. So we have a vertical line passing through the 0.-50. And we also have another vertical line passing through the .50. Both of these lines are solid due to the equal sign in our inequality. And then X being in the middle of these two values negative five and five, we have the area in between the two lines shaded in the graph. So now that we have graphed our first inequality, we can move on to graphing the second inequality that we found which if we recall is negative seven is less than or equal to Y which is less than or equal to seven. So graphing this inequality, we have two lines again, but now the first line is crossing crossing through the 20.0 and seven and the second line is passing through the 70.0 and negative seven. Both lines are solid because of the equal sign and the inequality and Y is in between negative seven and seven, which means the area in between the two lines is shaded. So now that we have graphed our two individual inequalities, we can graph the system altogether by combining our two graphs. So combining the two graphs again displays our system of inequalities. And looking at this graph, we have the same two horizontal lines, one line passing through the 10.0 and seven and the other line passing through the 70.0 negative seven. And then we have our two vertical lines, one line passing through 50 and the other line passing through negative 50, each of these lines are still solid because of the equal sign and the inequalities. But now with the new boundaries here, combining the boundaries, we now have the inside of these four lines shaded, the area bounded by these lines is shaded. So our solution is this graph but also the solid lines as well as the shaded line, the shaded area in between the lines. So with this graph, we are left with answer choice A which display displays the graph of our system of inequalities which are negative five is less than or equal to X, which is less than, or equal to five and negative seven is less than or equal to Y which is less than or equal to seven. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 69–70, rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. |x|≤2, |y|≤3

Key Concepts

Absolute Value Inequalities

Graphing Inequalities in Rectangular Coordinates

Systems of Inequalities

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