Graph each inequality. x + 2y ≤ 6

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Identify the inequality given: \(x + 2y \leq 6\). This represents a linear inequality in two variables.

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 \(y \leq \frac{6 - x}{2}\).

Graph the boundary line \(y = \frac{6 - x}{2}\). Since the inequality is \(\leq\), the boundary line should be solid, indicating points on the line satisfy the inequality.

Determine which side of the boundary line to shade by testing a point not on the line, such as the origin \((0,0)\). Substitute into the inequality: \(0 + 2(0) \leq 6\) which simplifies to \(0 \leq 6\), a true statement, so shade the region containing \((0,0)\).

Shade the region below or on the line \(y = \frac{6 - x}{2}\) to represent all solutions to the inequality \(x + 2y \leq 6\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 x + 2y ≤ 6, solving for y gives y ≤ (-1/2)x + 3. 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 the test point satisfies the inequality, shade the region containing that point; otherwise, shade the opposite side. This confirms the correct solution area.