/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Graph each inequality. $$x y \... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 22

Graph each inequality. $$x y \leq 1$$

Short Answer

Expert verified
Graph the boundary hyperbola \(xy=1\), test a point like (0,0) to find the region, and shade the region including the boundary.

Step by step solution

01

Graph the Boundary

First, graph the boundary of the inequality. The boundary equation is obtained by replacing the inequality symbol with an equals sign. In this case, the boundary is given by the equation: \[xy = 1\]Graph this equation on the coordinate plane by plotting several points (e.g., (1,1), (2, 0.5), (0.5, 2), etc.) and then drawing the hyperbola.
02

Determine the Inequality Region

To determine which region satisfies the inequality, pick a test point that is not on the boundary. A common test point is the origin (0,0). Substitute this point into the inequality: \[0 \times 0 less 1\]This inequality is true, so the region containing (0,0) satisfies the inequality.
03

Shade the Appropriate Region

Since the inequality \(xy \leq 1\) includes the boundary, the region where the test point (0,0) lies should be shaded. Shade the region underneath and to the right of the hyperbola down towards the axes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

boundary equation
When graphing an inequality, the first step is to identify the boundary equation. For the inequality \(xy \leq 1\), the boundary equation is found by replacing the inequality symbol with an equals sign: \(xy = 1\). The boundary shows the line or curve that separates the coordinate plane into different regions. For the given equation, the boundary forms a hyperbola. This hyperbola will help us determine the feasible region for our inequality.
test point
After drawing the boundary, we need to determine which region of the coordinate plane satisfies the inequality. To accomplish this, we use a test point. A commonly used point for simplicity is the origin (0,0). For our inequality \(xy \leq 1\), we substitute (0,0) into it:
\[0 \times 0 \leq 1\]
This inequality holds true, so the region that includes the origin is part of the solution. The test point helps us confirm if a region satisfies the inequality.
hyperbola
In our exercise, the boundary equation \(xy = 1\) forms a hyperbola. A hyperbola is a type of curve on the coordinate plane that appears as two symmetric branches. To graph it, plot points like (1,1), (2, 0.5), and (0.5, 2). Once several points are plotted, connect them smoothly to reveal the hyperbolic shape. This curve is essential for visualizing where the inequality's region lies.
coordinate plane
The coordinate plane is the stage where we graph our inequalities. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is defined by an ordered pair (x, y). For our inequality \(xy \leq 1\), plotting on this plane helps us distinguish between different regions. Ensure that you correctly position all points and the boundary hyperbola. Lastly, shade the appropriate region based on the test point to complete the graph.

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