Question

match the graph below to

1. $-3x + 2y > 4$

solutions:
( , )
( , )
graph:

Explanation:

Step1: Rewrite the inequality

First, we rewrite the inequality \(-3x + 2y>4\) in slope - intercept form (\(y = mx + b\)) to make it easier to analyze.
Add \(3x\) to both sides of the inequality: \(2y>3x + 4\).
Then divide both sides by 2: \(y>\frac{3}{2}x + 2\).

Step2: Analyze the boundary line

The boundary line of the inequality \(y>\frac{3}{2}x + 2\) is the line \(y=\frac{3}{2}x + 2\). Since the inequality is \(y>\frac{3}{2}x + 2\) (not \(y\geq\frac{3}{2}x + 2\)), the boundary line should be a dashed line (to indicate that the points on the line are not included in the solution set).

Step3: Find test points

We can choose test points to determine which side of the line to shade. A common test point is the origin \((0,0)\).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y>\frac{3}{2}x+2\):
\(0>\frac{3}{2}(0)+2\), which simplifies to \(0 > 2\). This is false. So the origin is not in the solution set. We should shade the side of the line that does not include the origin.

Step4: Find solution points

Let's find two points that satisfy the inequality \(y>\frac{3}{2}x + 2\).

• Let's take \(x = 0\). Then from the inequality \(y>\frac{3}{2}(0)+2=2\). We can choose \(y = 3\). So the point \((0,3)\) is a solution because \(3>\frac{3}{2}(0)+2\) (i.e., \(3 > 2\)).
• Let's take \(x = 2\). Then \(y>\frac{3}{2}(2)+2=3 + 2=5\). We can choose \(y = 6\). So the point \((2,6)\) is a solution because \(6>\frac{3}{2}(2)+2\) (i.e., \(6>5\)).

Answer:

Solution points: \((0,3)\), \((2,6)\) (answers may vary, any points satisfying \(y>\frac{3}{2}x + 2\) are correct). For the graph, it should be a dashed line with equation \(y=\frac{3}{2}x + 2\) and the region above the line shaded (since the origin is not in the solution set and we shade the side where \(y>\frac{3}{2}x + 2\)).