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Chapter 7: Problem 7

Draw an \(x-y\) coordinate frame and shade the region for which \(x<3\) and \(y>-2\).

Short Answer

Expert verified
Shade the region above \(y = -2\) and to the left of \(x = 3\) on the \(x-y\) plane.

Step by step solution

01

Identify Constraints

Given the inequalities, we need to identify the constraints: \(x < 3\) and \(y > -2\). These constraints represent the conditions for shading on the \(x-y\) plane.
02

Draw the Axes

Start by drawing the \(x-y\) coordinate frame. Label the horizontal line as the \(x\)-axis and the vertical line as the \(y\)-axis.
03

Plot the Boundary Lines

Plot the boundary lines for the inequalities. Draw a vertical dashed line at \(x = 3\) and a horizontal dashed line at \(y = -2\). These lines are dashed because the inequalities are strict (they do not include the line itself).
04

Shade the Appropriate Region

Shade the region that satisfies both inequalities. For \(x < 3\), shade to the left of the \(x = 3\) line. For \(y > -2\), shade above the \(y = -2\) line. The common region above \(y = -2\) and to the left of \(x = 3\) should be shaded.

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Key Concepts

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

Understanding Graphing Inequalities
Graphing inequalities involves using boundary lines to divide the coordinate plane into different regions. Each region reflects whether a particular area satisfies the given inequality:
  • For inequalities with the greater than symbol (> or 鈮�), we consider regions above the boundary if we're dealing with the y-axis or to the right if it involves the x-axis.
  • For inequalities that have less than (< or 鈮�), we consider regions below the boundary on the y-axis or to the left if it involves the x-axis.
  • When an inequality is strict, using (< or >), the boundary line is dashed to indicate it is not included in the solution set.
  • In the case of non-strict inequalities (鈮� or 鈮�), the boundary line is solid.
To effectively communicate solutions with graphing, plot both boundaries and identify where these shaded regions overlap. That way, you focus on the common area satisfying all criteria.
The Coordinate System as the Graphical Playground
The coordinate system is like a map that helps us find locations by using pairs of numbers, which are called coordinates.
The coordinate plane is divided into four quadrants by the x-axis and y-axis.
  • The x-axis is the horizontal line, while the y-axis is the vertical line.
  • Coordinates are written as (x, y), where x tells you how far to move right or left, and y tells you how far to move up or down.
  • Positive x-values move to the right of the origin (0,0), and negative x-values move to the left.
  • Similarly, positive y-values move upwards, while negative moves downwards.
By plotting lines or points, you can visualize and analyze mathematical relationships such as inequalities, which helps in understanding their solutions quicker with clarity.
Mastering Shading Regions
Shading regions is like painting the area where solutions lay in a coordinate plane. The goal is to highlight the area that satisfies the inequality's conditions.
  • After plotting the boundary lines for inequalities, determine which side of the line you need to shade. This step is crucial for capturing the correct solution area.
  • For the inequality x < 3, the shaded region will be on the left side of the line x = 3.
  • For y > -2, you will shade above the line y = -2.
  • The region where both shaded areas overlap is your solution where both inequalities are satisfied.
Visualizing these shaded regions can simplify complex problems, making it easier to see where solutions intersect.

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