Ggraph each inequality. x+2y≤8

Verified step by step guidance

1. Identify the inequality to graph: \(x + 2y \leq 8\).
2. Rewrite the inequality in slope-intercept form by isolating \(y\): subtract \(x\) from both sides to get \$2y \leq 8 - x\(, then divide both sides by 2 to obtain \)y \leq \frac{8 - x}{2}$.
3. Graph the boundary line \(y = \frac{8 - x}{2}\), which can be rewritten as \(y = 4 - \frac{1}{2}x\). This is a straight line with a y-intercept of 4 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 that points on the line satisfy the inequality.
5. Shade the region below the line because \(y\) is less than or equal to \$4 - \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

Rearranging the inequality into slope-intercept form (y = mx + b) helps in graphing. For example, x + 2y ≤ 8 can be rewritten as y ≤ -1/2 x + 4. This form clearly shows the slope and y-intercept, making it easier to plot the boundary line accurately.

Testing Points to Determine the Solution Region

After graphing the boundary line, select a test point not on the line (often the origin) to check if it satisfies the inequality. If it does, shade the region containing that point; if not, shade the opposite side. This step ensures the correct half-plane is identified as the solution set.