The left side of the equation must be less than or equal to 4 in order to satisfy the equation, so plugging in each of the values for x and y, we see:

The only point that satisfies the inequality is the point (-3,-2), since it yields an answer that is less than or equal to 4.

Recall the equation of circle:

 where r is the radius and the center of the circle is at (h,k).

The center of the circle is (-4,-3), so plugging those values in for x and y yields the response that 0 is less than or equal to 4, which is a true statement, so the center does satisfy the inequality. 

Check the center of the circle to see if that point satisfies the inequality. When evaluating the function at the center (-4,-3), we see that it does satisfy the equation, so it can be in the shaded region of the graph. Therefore the shading is inside of the circle.

The -intercepts of a function are the points where . When we substitute this into our equation, we get: 

. 

Adding nine to both sides, 

. 

Modifying the equation to get like bases get us,

Since .

Now we can set the exponents equal to eachother and solve for .

Thus, 

. 

Giving us our final solution: 

. 

The general equation of a circle is where the center is and the radius is .

In this case, the center is and the radius is , so the equation for this circle is .

The circle is shaded on the inside, which means that choosing any point  and plugging it in for would produce something less than .

Therefore, our answer is .

The general equation for a circle is where the center is and its radius is .

In this case, the center is  and the radius is , so the equation for the circle is .

We can simplify this equation to: .

The circle is shaded on the inside, which means that choosing any point and plugging it in for  would produce something less than . 

Therefore, our answer is .

The left side of the equation must be greater than or equal to 25 in order to satisfy the equation, so plugging in each of the values for x and y, we see that:

The only point that satisfies the inequality is (7,4) since it yields an answer that is greater than or equal to 25.

The hyperbola in the question's graph is .

This could be discerned because it subtracts from and not the other way around, putting the zeros on the  axis. This narrows the answers down to and .

Test a point to tell which inequality is being used in the graph.

The point  is within the shaded region, for example. 

, which is greater than 16, so the answer is .

The hyperbola in the graph has y-intercepts rather than x-intercepts, so the equation must be in the form and not the other way around.

The y-intercepts are at 1 and -1, so the correct equation will have just and not .

The answer not must either be,

  or .

To see which, test a point in the shaded area.

For example, . 

, which is less than 1, so the answer is .

The equation for a horizontal hyperbola is . The equation for a vertical hyperbola is . In both,  is the center of the hyperbola. Hyperbolic inequalities use an inequality sign rather than an equals sign, but otherwise have the same form as hyperbolic equations. The graph shows a vertical hyperbola, so in its corresponding inequality the y-term must appear first. The center is shaded, so the left side of the graph’s corresponding inequality (the side containing the variables x and y) is less than the constant on the right side. The lines are dashed rather than solid, so the inequality sign must be  rather than .