How do I graph the solution sets of the inequalities xy ≤ 4?

To graph the solution set of the inequality xy ≤ 4, we start by rewriting it in a more usable form. We can express it as:

y ≤ 4/x

Next, we need to consider the equation y = 4/x. This represents a hyperbola. To graph the inequality, follow these steps:

1. Identify the intercepts: The curve has vertical and horizontal asymptotes at x = 0 and y = 0, which means it does not touch these axes.
2. Plot points: Evaluate the function for different values. For example:
• If x = 1, then y = 4.
• If x = 2, then y = 2.
• If x = 4, then y = 1.
• If x = -1, then y = -4.
• If x = -2, then y = -2.
3. Draw the hyperbola: Connect these points smoothly to sketch the hyperbola. Since we are interested in y ≤ 4/x, we will shade the area below the curve.

In conclusion, the solution set includes all the points on or below the curve y = 4/x, and does not include the x or y axes. Make sure to indicate the shaded area clearly on your graph!