Graph each inequality. x + 2y ≤ 6

Verified step by step guidance

1. Identify the inequality to graph: \(x + 2y \leq 6\).
2. Rewrite the inequality in slope-intercept form by isolating \(y\): subtract \(x\) from both sides to get \$2y \leq 6 - x\(, then divide both sides by 2 to obtain \)y \leq \frac{6 - x}{2}$.
3. Graph the boundary line \(y = \frac{6 - x}{2}\), which can be rewritten as \(y = 3 - \frac{1}{2}x\). This is a straight line with a y-intercept of 3 and a slope of \(-\frac{1}{2}\).
4. Since the inequality is \(\leq\) (less than or equal to), draw the boundary line as a solid line to indicate points on the line satisfy the inequality.
5. Shade the region below the line because \(y\) is less than or equal to \$3 - \frac{1}{2}x\(. You can test a point not on the line, such as \)(0,0)$, to confirm which side to shade by substituting into the inequality.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves first graphing the related linear equation as a boundary line. The inequality symbol determines whether the boundary is solid (≤ or ≥) or dashed (< or >). The solution region is the set of points that satisfy the inequality, typically shaded on one side of the boundary line.

Slope-Intercept Form of a Line

The slope-intercept form, y = mx + b, expresses a line where m is the slope and b is the y-intercept. Converting the inequality into this form helps identify the boundary line and makes graphing easier by showing how y changes with x.

Testing Points to Determine the Solution Region

After graphing the boundary line, a test point (often (0,0) if not on the line) is substituted into the inequality to check if it satisfies the inequality. If true, the region containing the test point is shaded; if false, the opposite side is shaded.