To graph the inequality [tex]\(-3x + 2y < 4\)[/tex], we need to follow a series of steps to understand the boundary line and the region that satisfies the inequality.
### Step 1: Convert the Inequality to an Equation
First, convert the inequality [tex]\(-3x + 2y < 4\)[/tex] to an equation:
[tex]\[
-3x + 2y = 4
\][/tex]
This allows us to determine the boundary line.
### Step 2: Find the Equation of the Boundary Line
Solve the equation for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[
2y = 3x + 4
\][/tex]
[tex]\[
y = \frac{3}{2}x + 2
\][/tex]
This is the equation of the line that will serve as the boundary for the inequality.
### Step 3: Plot the Boundary Line
To plot this line, we'll find two points on the line:
1. When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = \frac{3}{2}(0) + 2 = 2
\][/tex]
So, one point is [tex]\((0, 2)\)[/tex].
2. When [tex]\( y = 0 \)[/tex]:
Solve [tex]\( 0 = \frac{3}{2}x + 2 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
0 = \frac{3}{2}x + 2
\][/tex]
[tex]\[
\frac{3}{2}x = -2
\][/tex]
[tex]\[
x = -\frac{4}{3}
\][/tex]
So, another point is [tex]\(\left(-\frac{4}{3}, 0\right)\)[/tex].
Plot these points on the Cartesian plane and draw a straight line through them. This line represents the equation [tex]\(-3x + 2y = 4\)[/tex].
### Step 4: Determine the Region to Shade
Since the inequality is [tex]\(-3x + 2y < 4\)[/tex], we need to shade the region below the line, but let's confirm with a test point:
- Choose a test point that is not on the line. A common choice is the origin [tex]\((0, 0)\)[/tex].
Substitute [tex]\( (0, 0) \)[/tex] into the inequality:
[tex]\[
-3(0) + 2(0) < 4
\][/tex]
[tex]\[
0 < 4
\][/tex]
This is true, so the region that satisfies the inequality includes the test point [tex]\((0, 0)\)[/tex], indicating that we need to shade the region below the line.
### Step 5: Graph the Inequality
- Draw the boundary line [tex]\( y = \frac{3}{2}x + 2 \)[/tex]. Since the inequality is strict ([tex]\(<\)[/tex]), use a dashed line to indicate that points on the line itself are not included.
- Shade the region below the dashed line to represent all the points [tex]\((x, y)\)[/tex] that satisfy [tex]\(-3x + 2y < 4\)[/tex].
### Summary
Here’s a step-by-step guide to graph the inequality:
1. Graph the boundary line [tex]\( y = \frac{3}{2}x + 2 \)[/tex] as a dashed line.
2. Choose a test point (like the origin) to determine which side of the line to shade.
3. Shade the region below the boundary line, as that represents the set of points that satisfy [tex]\(-3x + 2y < 4\)[/tex].
By following these steps, you should have a complete graph of the inequality [tex]\(-3x + 2y < 4\)[/tex].
