To graph the inequality [tex]\( 2x + 3y > -3 \)[/tex], follow these steps:
### Step 1: Rewrite the Inequality as an Equation
Firstly, convert the inequality into an equation to find the boundary line. The equation corresponding to the inequality [tex]\( 2x + 3y > -3 \)[/tex] is:
[tex]\[ 2x + 3y = -3 \][/tex]
### Step 2: Find the Intercepts
To graph the line, determine the x-intercept and y-intercept.
- Y-intercept: Set [tex]\( x = 0 \)[/tex] to find the y-intercept.
[tex]\[
2(0) + 3y = -3
\][/tex]
[tex]\[
3y = -3
\][/tex]
[tex]\[
y = -1
\][/tex]
So, the y-intercept is [tex]\((0, -1)\)[/tex].
- X-intercept: Set [tex]\( y = 0 \)[/tex] to find the x-intercept.
[tex]\[
2x + 3(0) = -3
\][/tex]
[tex]\[
2x = -3
\][/tex]
[tex]\[
x = -\frac{3}{2}
\][/tex]
So, the x-intercept is [tex]\((-\frac{3}{2}, 0)\)[/tex].
### Step 3: Draw the Boundary Line
Plot the intercepts [tex]\((0, -1)\)[/tex] and [tex]\((-\frac{3}{2}, 0)\)[/tex] on the coordinate plane, and draw a straight line through these points. This line represents the equation [tex]\( 2x + 3y = -3 \)[/tex].
### Step 4: Determine the Shaded Region
Because the inequality is [tex]\( 2x + 3y > -3 \)[/tex], we need to determine which side of the line represents the solution set.
- Test a Point: Choose a test point not on the line. A convenient choice is the origin [tex]\((0,0)\)[/tex].
[tex]\[
2(0) + 3(0) > -3
\][/tex]
[tex]\[
0 > -3
\][/tex]
This is true, so the region that includes the origin satisfies the inequality.
### Step 5: Shade the Correct Region
Shade the half-plane that includes the origin [tex]\((0,0)\)[/tex].
### Step 6: Boundary Line Type
Since the inequality is strictly greater than [tex]\((>)\)[/tex], the boundary line [tex]\(2x + 3y = -3\)[/tex] should be dashed, indicating points on the line are not included in the solution set.
In conclusion, the graph of the inequality [tex]\(2x + 3y > -3\)[/tex] is a dashed line representing [tex]\(2x + 3y = -3\)[/tex], with the region above the line (including the origin) shaded.
