Question

graph the inequality on the axes below.
3x - 2y ≤ 4

Explanation:

Step1: Convert to slope - intercept form

Start with the inequality \(3x - 2y\leq4\). We want to solve for \(y\).
First, subtract \(3x\) from both sides: \(- 2y\leq - 3x + 4\).
Then, divide each term by \(-2\). Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes. So we get \(y\geq\frac{3}{2}x - 2\).

Step2: Graph the boundary line

The boundary line is \(y=\frac{3}{2}x - 2\). This is a linear equation in slope - intercept form \(y = mx + b\), where the slope \(m=\frac{3}{2}\) and the \(y\) - intercept \(b=-2\).

• Plot the \(y\) - intercept: The point \((0,-2)\) is on the line.
• Use the slope to find another point. The slope \(\frac{3}{2}\) means we can go up 3 units and then right 2 units from the \(y\) - intercept. So from \((0,-2)\), moving up 3 units gives \(y=-2 + 3=1\) and moving right 2 units gives \(x = 0+2 = 2\). So the point \((2,1)\) is also on the line.
• Since the inequality is \(y\geq\frac{3}{2}x - 2\) (the "greater than or equal to" sign), the boundary line should be a solid line (because the points on the line are included in the solution set).

Step3: Shade the region

To determine which side of the line to shade, we can test a point that is not on the line. A common test point is the origin \((0,0)\).
Substitute \(x = 0\) and \(y = 0\) into the original inequality \(3x-2y\leq4\):
\(3(0)-2(0)=0\leq4\), which is true.
Since the origin satisfies the inequality, we shade the region that includes the origin. This region is above the line \(y=\frac{3}{2}x - 2\) (because \(y\geq\frac{3}{2}x - 2\) and the origin is above the line \(y=\frac{3}{2}x - 2\) when \(x = 0\)).

(Note: Since the problem asks to graph the inequality, the final answer is the graph with the solid line \(y=\frac{3}{2}x - 2\) and the region above the line shaded. If we were to describe the key features: the boundary line has a slope of \(\frac{3}{2}\), \(y\) - intercept at \((0, - 2)\), is a solid line, and the shaded region is above the line.)

Answer:

The graph has a solid boundary line \(y = \frac{3}{2}x-2\) (with \(y\) - intercept \((0,-2)\) and another point \((2,1)\) on the line) and the region above the line is shaded.