To graph the linear inequality [tex]\(2x - 3y < 12\)[/tex], follow these steps carefully:
1. Rewrite the Inequality as an Equation:
[tex]\[ 2x - 3y < 12 \][/tex]
First, let's consider the related equation [tex]\(2x - 3y = 12\)[/tex]. This will be our boundary line.
2. Find the Intercepts:
- X-intercept: Set [tex]\( y = 0 \)[/tex] in the equation [tex]\(2x - 3y = 12\)[/tex]:
[tex]\[ 2x - 3(0) = 12 \][/tex]
[tex]\[ 2x = 12 \][/tex]
[tex]\[ x = 6 \][/tex]
So, the x-intercept is [tex]\((6, 0)\)[/tex].
- Y-intercept: Set [tex]\( x = 0 \)[/tex] in the equation [tex]\(2x - 3y = 12\)[/tex]:
[tex]\[ 2(0) - 3y = 12 \][/tex]
[tex]\[ -3y = 12 \][/tex]
[tex]\[ y = -4 \][/tex]
So, the y-intercept is [tex]\((0, -4)\)[/tex].
3. Draw the Boundary Line:
Plot the line passing through the points [tex]\((6, 0)\)[/tex] and [tex]\((0, -4)\)[/tex]. This is the equation [tex]\(2x - 3y = 12\)[/tex]. Use a dashed line to plot this because the inequality is strict ([tex]\(<\)[/tex]), not inclusive ([tex]\(\leq\)[/tex]).
4. Determine the Shading Region:
To determine which side of the line to shade, select a test point that is not on the line. The point [tex]\((0, 0)\)[/tex] is an easy choice:
[tex]\[ 2(0) - 3(0) < 12 \][/tex]
[tex]\[ 0 < 12 \][/tex]
This statement is true, indicating that the point [tex]\((0, 0)\)[/tex] is part of the solution set. Hence, you will shade the half-plane that contains the origin.
5. Shade the Correct Region:
Shade the region on the side of the line that includes the origin. This region represents all the points [tex]\((x, y)\)[/tex] that satisfy the inequality [tex]\(2x - 3y < 12\)[/tex].
6. Label the Axes and Add Details:
- Label the x-axis and y-axis appropriately.
- Add a title "Graph of [tex]\(2x - 3y < 12\)[/tex]".
### Graph Summary:
- Draw a dashed line through points [tex]\((6, 0)\)[/tex] and [tex]\((0, -4)\)[/tex].
- Shade the region below this line (since [tex]\(2x - 3y < 12\)[/tex] holds there).
