Graph each inequality. 4y - 3x ≤ 5

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Rewrite the inequality in terms of $y$ to make it easier to graph. Start with the given inequality: $4y - 3x \leq 5$.

Add \$3x$ to both sides to isolate the $y$ term: $4y \leq 3x + 5$.

Divide every term by 4 to solve for $y$: $y \leq \frac{3}{4}x + \frac{5}{4}$.

Graph the boundary line $y = \frac{3}{4}x + \frac{5}{4}$. Since the inequality is $\leq$, use a solid line to indicate that points on the line satisfy the inequality.

Shade the region below the line because $y$ is less than or equal to the expression on the right side. This shaded area represents all solutions to the inequality.

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주요 개념

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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 half-plane where the inequality holds true, which is shaded on the graph.

Rearranging Inequalities into Slope-Intercept Form

To graph the inequality easily, rewrite it in slope-intercept form (y = mx + b). This involves isolating y on one side, which helps identify the slope and y-intercept, making it straightforward to plot the boundary line and determine the shading direction.

Testing Points to Determine the Solution Region

After graphing the boundary line, select a test point not on the line (commonly (0,0)) to check if it satisfies the inequality. If it does, shade the region containing that point; if not, shade the opposite side. This confirms the correct half-plane representing the solution set.