Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Inequalities
Linear inequalities are mathematical expressions that involve a linear function and an inequality sign (such as ≤, ≥, <, or >). They represent a range of values rather than a single solution, indicating that the values of the variable can be less than or equal to (or greater than or equal to) a certain expression. Understanding how to interpret and graph these inequalities is essential for visualizing the solution set.
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Graphing Linear Equations
Graphing linear equations involves plotting points on a coordinate plane that satisfy the equation. The equation y = mx + b represents a line where 'm' is the slope and 'b' is the y-intercept. For the inequality y ≤ (-1/2)x + 2, one must first graph the line y = (-1/2)x + 2 as a dashed line (since the inequality is not strict) and then shade the region below the line to represent all the points that satisfy the inequality.
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Shading Regions in Graphs
Shading regions in graphs is a technique used to indicate the solution set of an inequality. For a linear inequality like y ≤ (-1/2)x + 2, the area below the line is shaded to show that all points in this region satisfy the inequality. This visual representation helps in understanding which values of x and y are included in the solution set, making it easier to analyze and interpret the results.
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