Question

which graph represents the inequality ( 2x - 3y < 6 )?
options: a, b, c, d (with corresponding graphs)

Explanation:

Step1: Rewrite the inequality

First, we start with the inequality \(2x - 3y<6\). Let's rewrite it in slope - intercept form (\(y = mx + b\)) to analyze the boundary line and the shaded region.
Subtract \(2x\) from both sides: \(- 3y<-2x + 6\).
Divide both sides by \(-3\). Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes. So we get \(y>\frac{2}{3}x - 2\).

Step2: Analyze the boundary line

The equation of the boundary line is \(y=\frac{2}{3}x - 2\). The slope \(m = \frac{2}{3}\) (positive, so the line should be increasing) and the \(y\) - intercept \(b=-2\) (the line crosses the \(y\) - axis at \((0, - 2)\)). Also, since the inequality is \(y>\frac{2}{3}x - 2\), the boundary line should be dashed (because the inequality is strict, \(>\) not \(\geq\)) and we shade the region above the line.

Step3: Analyze each option

• Option A: The shaded region is below the line, and the inequality sign in the shaded region would correspond to \(y <\) something, so this is incorrect.
• Option B: The boundary line is dashed (good, since the inequality is strict), the slope is positive (the line is increasing), the \(y\) - intercept is \(-2\), and the shaded region is above the line (corresponding to \(y>\frac{2}{3}x - 2\)). Let's check the intercepts. For the line \(y=\frac{2}{3}x - 2\), when \(x = 0\), \(y=-2\); when \(y = 0\), \(\frac{2}{3}x-2=0\Rightarrow\frac{2}{3}x=2\Rightarrow x = 3\). The line in option B passes through \((0,-2)\) and \((3,0)\), and the shaded region is above the line, which matches \(y>\frac{2}{3}x - 2\) (or \(2x-3y < 6\) after re - arranging).
• Option C: The \(y\) - intercept of the line in this option does not seem to be \(-2\), so this is incorrect.
• Option D: The slope of the line in this option does not seem to be \(\frac{2}{3}\), and the shaded region is also not correct.

Answer:

B