Shading is a visual technique used to represent the intensity or depth of an object or surface within a mathematical or graphical context. It involves the application of varying degrees of darkness or color to create the illusion of three-dimensionality, volume, and texture.
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Shading is used to visually represent the solution set of a linear inequality or an absolute value inequality on a coordinate plane.
The shaded region on a graph indicates the set of all points that satisfy the given inequality or absolute value inequality.
The type of inequality (greater than, less than, greater than or equal to, less than or equal to) determines the direction of the shaded region.
Absolute value inequalities result in two distinct shaded regions, one on either side of the vertical line defined by the absolute value expression.
The shaded region can be used to analyze the behavior and properties of the inequality, such as the range of values that satisfy the inequality.
Review Questions
Explain how the type of inequality (greater than, less than, etc.) affects the direction of the shaded region on a graph.
The type of inequality determines the direction of the shaded region on a graph. For example, a linear inequality of the form $y > mx + b$ would result in a region above the line, while a linear inequality of the form $y ≤ mx + b$ would result in a region below the line. Similarly, an absolute value inequality of the form $|x - a| < b$ would produce two shaded regions, one on either side of the vertical line $x = a$, representing the set of values that are within a distance of $b$ from $a$.
Describe the relationship between shading and the solution set of an inequality or absolute value inequality.
The shaded region on a graph represents the solution set of the inequality or absolute value inequality. All points within the shaded region satisfy the given inequality, while points outside the shaded region do not. The shaded region can be used to analyze the range of values that satisfy the inequality, as well as the behavior and properties of the inequality, such as the intervals where the inequality is true or false.
Analyze how the shading of an absolute value inequality differs from the shading of a linear inequality, and explain the significance of this difference.
The shading of an absolute value inequality differs from the shading of a linear inequality in that absolute value inequalities result in two distinct shaded regions, one on either side of the vertical line defined by the absolute value expression. This is because absolute value inequalities involve the distance of a value from a specific point, rather than a comparison to a single line. The two shaded regions represent the set of values that are within a certain distance of the point defined by the absolute value expression. This distinction in shading is significant because it allows for the analysis of the behavior and properties of absolute value inequalities, which can be more complex than linear inequalities.
A mathematical statement that compares two values using an inequality symbol, such as greater than (>) , less than (<) , greater than or equal to (≥) , or less than or equal to (≤) .