Question

for a chemistry demonstration, malik is mixing sodium hydroxide with vinegar. the relationship between the amount of sodium hydroxide, x, and the amount of vinegar, y, is graphed below. which inequality best represents this graph? a 4x - 6y < 0 b 4x - 6y ≥ 0 c 4x - 6y > 0 d 4x - 6y ≤ 0

Explanation:

Step1: Find the equation of the boundary - line

The boundary - line passes through the points \((0,6)\) and \((9,0)\). The slope \(m\) of the line using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) is \(m=\frac{0 - 6}{9 - 0}=-\frac{2}{3}\). Using the slope - intercept form \(y=mx + b\) with \(b = 6\), the equation of the line is \(y=-\frac{2}{3}x+6\), which can be rewritten as \(2x+3y = 18\) or \(4x + 6y=36\).

Step2: Determine the inequality sign

The region is shaded below the line and the line is solid. For a line \(Ax+By = C\), if the region below the solid line is shaded, the inequality is \(Ax+By\leq C\). So the inequality is \(4x + 6y\leq36\) or \(4x+6y - 36\leq0\), which is equivalent to \(4x+6y\leq0\) (assuming the non - negative quadrant is considered, and the general form of the inequality for the shaded region).

Answer:

D. \(4x+6y\leq0\)