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^2+y^2<16, y≥2^x

Graph of the system of inequalities: x^2 + y^2 < 25 and y ≥ e^(x/2).

Accepted Answer

Hello, today we're gonna be solving the system of inequalities by using the method of graphing. So what we are given is a system of two inequalities. The best way to solve this system of inequalities is to analyze the graphs individually. So let's go ahead and graph each inequality first. If we take a look at the first inequality, we are given X squared plus Y squared is less than 25. The easiest way to graph. This is to imagine this as the equation of a circle. If we replace the inequality sign with an equal symbol, we get X squared plus Y squared is equal to 25. This matches the format of the equation of a circle. And recall that the general form of the equation of a circle is X squared plus Y squared is equal to R squared. So far in equality, we can graph it as a circle here R squared is equal to 25. And if we take the square root of both sides, we get R is equal to five. That means that this inequality represents the graph of a circle with a radius of five. So if we create an axis for this inequality, we're going to have a circle of radius of five. So the end points are gonna be located at five, comma zero zero, comma five negative five comma zero and zero comma negative five. Now, the question is, do we want to include the values that are on the circle itself? Well, this inequality is using the less end symbol and this means that we don't want to include any values that lie on the actual graph of the circle. So we're going to graph the circle using a dotted line since we are not including any values that lie directly on the circle. And now we need to include the region of solutions. Do we want to include the solutions that are in the region outside of the circle or do we want to include the region inside the circle? But what our inequality is saying is that any X and Y values that we get need to be less than the radius. And the only values that are less than the radius are gonna be contained inside of the circle. So that means we want to include the set of solutions that exist inside of the circle. And this is going to be the graph for our first inequality. Now let's go ahead and graph our second inequality. The second inequality is given to us as Y greater than or equal to E to the power of X over two. Now this is the graph of an exponential function. But the one common thing that every exponential function has is that they have intercepting point on the Y axis at zero. Comma one furthermore, the range is always gonna be slightly above the X axis and continue to infinity. So this is going to be the graph of the exponential function. And we graphed it using a solid line because the inequality symbol given to us is greater than or equal to. That means that we want to include the values that are located on the exponential function itself. Now the question is, do we want to shape the region that is below the exponential function or shape the region that is above the exponential function? Well, the easiest way to determine that is to take a testing point. The easiest testing point to pick in this case is going to be 00. If we plug it into the inequality and it makes it a true inequality, then we're gonna shade the region below the graph. Otherwise we want to include the region above the graph. So if we plug in 00 into an inequality, we get zero is greater than equal E to the power of 0/2 0/2 is going to reduce to give a zero. So we have zero is greater than E to the power of zero. Any value raised to the power of zero is going to give us one. So inequality is going to simplify to give us zero greater than equal to positive one. But this is a false statement which means that we don't want to include the region below the graph. We want to include the region above the exponential graph. So now that we have the shape of both of the inequalities, we want to go ahead and combine the graphs into a single axis and take a look at the overlapping sections. So if we draw another axis recall that the first inequality was the graph of a circle with a radius of five. And we also graphed it using a dotted line because we didn't want to include any values that lie on the actual circle. Next, for the second inequality, we had a common point at zero comma one and it was an exponential graph that had a rough shape out like this. Now we know that the entirety of the circle was shaded at least the inner part is. But if we shape the top part of the exponential graph, we're going to get an overlapping section of the two graphs which creates this wedge like shape. And so that means that this intersection intersecting region of the two graphs is going to be our solution. And the option that accurately represents this region of intersection is going to be B if we take a look at B B has a circle with a rough radius of five and it contains the exponential function where it starts at zero comma one and the overlapping section matches the one that we came up with earlier. Therefore, the answer to this problem is going to be B 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^2+y^2<16, y≥2^x

Key Concepts

Inequalities

Graphing Systems of Inequalities

Exponential Functions

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