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 .

The equation for a horizontal hyperbola is. The equation for a vertical hyperbola is . Hyperbolic inequalities use an inequality sign rather than an equals sign, but otherwise have the same form as hyperbolic equations. The fact that the right side of the inequality is not equal to 1 does not change the fact that ,  and  represent hyperbolas, since THESE can all be simplified to create an inequality with 1 on the right side (by dividing both sides of the equation by the constant on the right side of the inequality.) Answer choice  is the only option in which the two terms on the left side of the inequality are combined using addition rather than subtraction, creating an ellipse rather than a hyperbola. (The equation for an ellipse is .)

 The equation for a horizontal hyperbola is . The equation for a vertical hyperbola is . Hyperbolic inequalities use an inequality sign rather than an equals sign, but otherwise have the same form as hyperbolic equations. The presence of coefficients in  and  does not change the fact that  and  represent hyperbolas, since both can be simplified to remove those coefficients (by dividing the numerator and denominator of terms with coefficients by those coefficients.) Answer choice  is missing an exponent of 2 on the first term in the inequality, and therefore does not match the form of a hyperbola. 

The equation for a horizontal hyperbola is . The equation for a vertical hyperbola is . In both, (h, v) 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 horizontal hyperbola, so in its corresponding inequality the x-term must appear first. The center is not shaded, so the left side of the graph’s corresponding inequality (the side containing the variables x and y) is greater than the constant on the right side. The lines are dashed rather than solid, so the inequality sign must be  rather than . The center lies at (-1, 1), so x must be followed by the constant 1, and y must be followed by the constant -1.

The equation for a horizontal hyperbola is . The equation for a vertical hyperbola is . In both, (h, v) 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 horizontal hyperbola, so in its corresponding inequality the x-term must appear first. The center is not shaded, so the left side of the graph’s corresponding inequality (the side containing the variables x and y) is greater than the constant on the right side.

The equation for a horizontal hyperbola is . The equation for a vertical hyperbola is . In both, (h, v) 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 (0, 0), so neither x nor y can be followed by a constant. 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 solid rather than dashed, so the inequality sign must be  rather than .

The equation for a horizontal hyperbola is. The equation for a vertical hyperbola is . In both, (h, v) 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. Neither the x nor the y in the inequality are followed by constants, so the graph must be centered on (0,0). The x-term appears first, so the graph must show a horizontal hyperbola. The inequality sign is  rather than , so the lines must be dashed rather than solid. The left side is less than rather than greater than the constant, so the center must be shaded.