Graph the solution set of each inequality. 6 x-2 y ≤24 |...

Key Concepts

Half-Plane

Once the boundary line is drawn, the solution to the inequality is depicted as one of the two regions (half-planes) divided by the line. Determining which half-plane to shade can be done by selecting a test point not on the boundary line and checking whether it satisfies the original inequality.

Slope-Intercept Form

Converting a linear equation into slope-intercept form (y = mx + b) simplifies the graphing process. This form clearly shows the slope of the line and the y-intercept, making it straightforward to graph the boundary line by plotting the intercept and using the slope to determine another point.

Linear Inequality

A linear inequality in two variables involves an inequality relation (such as ?, <, ?, or >) between two linear expressions. It represents a range of solutions rather than a single solution, which when graphed, usually forms a half-plane on the coordinate system.

Graphing Linear Inequalities

Graphing a linear inequality requires first graphing the corresponding boundary line—obtained by converting the inequality to an equation—and then shading the region of the graph that satisfies the inequality. This process helps visualize all possible solutions that meet the inequality condition.

Boundary Line

The boundary line is the graph of the equation formed by replacing the inequality symbol with an equality. It delineates the edge of the solution set. For inequalities involving ? or ?, the boundary line is drawn solid to indicate that points on the line are included in the solution set.

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