In Exercises 39–45, graph each inequality. y ≤ (-1/2)x + 2

Verified step by step guidance

1. Step 1: Begin by identifying the inequality given: y ≤ (-1/2)x + 2. This represents a linear inequality where the boundary line is y = (-1/2)x + 2.
2. Step 2: Graph the boundary line y = (-1/2)x + 2. To do this, find the y-intercept (where x = 0). Substitute x = 0 into the equation to get y = 2. Plot the point (0, 2) on the graph.
3. Step 3: Determine the slope of the line, which is -1/2. This means for every 1 unit increase in x, y decreases by 1/2. Use this slope to plot another point starting from the y-intercept. For example, move 1 unit to the right (x = 1) and 1/2 unit down (y = 1.5). Plot this point.
4. Step 4: Draw the boundary line through the points. Since the inequality is ≤, the boundary line should be solid, indicating that points on the line are included in the solution.
5. Step 5: Shade the region below the line because the inequality is y ≤ (-1/2)x + 2. This represents all points where y is less than or equal to (-1/2)x + 2. The shaded region is the solution to the inequality.

Key Concepts

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 dependent variable (y) is less than or equal to a linear expression in terms of the independent variable (x). Understanding how to interpret and graph these inequalities is crucial for visualizing the solution set.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane that satisfy the equation. For the inequality y ≤ (-1/2)x + 2, first, the corresponding linear equation y = (-1/2)x + 2 is graphed as a straight line. The slope of -1/2 indicates a downward trend, and the y-intercept of 2 shows where the line crosses the y-axis. This foundational skill is essential for understanding how to represent inequalities graphically.

Shading the Solution Region

When graphing a linear inequality, the solution region is indicated by shading the area of the graph that satisfies the inequality. For y ≤ (-1/2)x + 2, the area below the line (including the line itself, since it is ≤) represents all the points (x, y) that fulfill the inequality. This visual representation helps in identifying all possible solutions and is a key aspect of understanding inequalities in a graphical context.