Answer:
The question you've shared is asking which inequality will have a shaded area below the boundary line when graphed on a coordinate plane.
Inequalities can be graphed on a coordinate plane, and the region representing the solution to the inequality is typically shaded. For an inequality in two variables (such as \( x \) and \( y \)), the boundary line is drawn by graphing the corresponding equation (the inequality becomes an equation when the inequality symbol is replaced with an equals sign).
The shaded region for the inequality will be:
- Below the boundary line if the inequality is of the form \( y < mx + b \) or \( y \leq mx + b \).
- Above the boundary line if the inequality is of the form \( y > mx + b \) or \( y \geq mx + b \).
Looking at the options:
A. \( y - x > 5 \): The boundary line
is \( y - x = 5 \), and the shaded area would be above the line because the inequality symbol is "greater than" (>).
B. \( 2x - 3y \leq 3 \): If we solve for \( y \), the inequality becomes \( y \geq \frac{2}{3}x - 1 \). The boundary line is \( y = \frac{2}{3}x - 1 \), and the shaded area would be above the line because the inequality symbol is "greater than or equal to" (\(\geq\)) when solved for \( y \).
C. \( 2x - 3y \): This is not an inequality, it's an equation.
D. \( 7x + 2y \leq 2 \): If we solve for \( y \), the inequality becomes \( y \leq -\frac{7}{2}x + 1 \). The boundary line is \( y = -\frac{7}{2}x + 1 \), and the shaded area would be below the line because the inequality symbol is "less than or equal to" (\(\leq\)).
E. \( 3x + 4y > 12 \): If we solve for \( y \), the inequality becomes \( y
> -\frac{3}{4}x + 3 \). The boundary line is \( y = -\frac{3}{4}x + 3 \), and the shaded area would be above the line because the inequality symbol is "greater than" (>).
So, the inequality that will have a shaded area below the boundary line is option D, \( 7x + 2y \leq 2 \). When this inequality is graphed, the shading will be on the side of the line where \( y \) values are less than or equal to the line \( y = -\frac{7}{2}x + 1 \).
Step-by-step explanation:
