Question

1 multiple choice 1 point which graph represents the inequality: 2x - 3y ≤ 9? (graphs of coordinate planes with shaded regions and lines are shown)

Explanation:

Step1: Rewrite the inequality

First, we rewrite the inequality \(2x - 3y\leq9\) in slope - intercept form (\(y = mx + b\) form).
Subtract \(2x\) from both sides: \(-3y\leq - 2x + 9\).
Divide both sides by \(-3\). When we divide an inequality by a negative number, the direction of the inequality sign changes. So we get \(y\geq\frac{2}{3}x - 3\).

Step2: Analyze the boundary line

The boundary line of the inequality \(y\geq\frac{2}{3}x - 3\) has a slope of \(\frac{2}{3}\) (positive slope) and a \(y\) - intercept of \(- 3\). Also, since the inequality is \(\geq\), the boundary line should be a solid line (because the inequality includes equality).

Step3: Analyze the shading

For the inequality \(y\geq\frac{2}{3}x - 3\), we shade the region above the line (because \(y\) is greater than or equal to the expression \(\frac{2}{3}x - 3\)).

Now let's analyze the given graphs:

• The first graph: The line has a negative slope, so it does not match our boundary line (our boundary line has a positive slope of \(\frac{2}{3}\)).
• The second graph: The line has a negative slope, so it does not match our boundary line.
• The third graph: The line has a positive slope (matches \(\frac{2}{3}\)), the \(y\) - intercept seems to be \(-3\) (when \(x = 0\), \(y=-3\)), and the shading is above the line (matches \(y\geq\frac{2}{3}x - 3\)).

Answer:

The third graph (the one with the positive - sloped line and shading above the line)