To graph the linear inequality [tex]\(2x - 3y < 12\)[/tex], follow these steps:
1. Convert the inequality to an equation:
[tex]\[
2x - 3y = 12
\][/tex]
This represents the boundary line of the inequality.
2. Find the intercepts of the boundary line:
- When [tex]\(x = 0\)[/tex]:
[tex]\[
2(0) - 3y = 12 \implies -3y = 12 \implies y = -4
\][/tex]
So the y-intercept is [tex]\((0, -4)\)[/tex].
- When [tex]\(y = 0\)[/tex]:
[tex]\[
2x - 3(0) = 12 \implies 2x = 12 \implies x = 6
\][/tex]
So the x-intercept is [tex]\((6, 0)\)[/tex].
3. Plot the boundary line:
Plot the points [tex]\((0, -4)\)[/tex] and [tex]\((6, 0)\)[/tex] on the coordinate plane and draw a straight line through these points.
4. Determine which side of the line to shade for the inequality:
Since the inequality is [tex]\(2x - 3y < 12\)[/tex], we need to determine which side of the line satisfies this inequality. A simple way to do this is to test a point that is not on the line, typically the origin [tex]\((0,0)\)[/tex]:
Substitute [tex]\((0,0)\)[/tex] into the inequality:
[tex]\[
2(0) - 3(0) < 12 \implies 0 < 12
\][/tex]
This is true, so the region that contains the origin is the solution to the inequality.
5. Shade the solution region:
Shade the area below the line [tex]\(2x - 3y = 12\)[/tex]. This represents all the points [tex]\((x, y)\)[/tex] that satisfy the inequality [tex]\(2x - 3y < 12\)[/tex].
6. Draw the graph:
On a coordinate plane:
- Plot the boundary line using [tex]\((0, -4)\)[/tex] and [tex]\((6, 0)\)[/tex].
- Shade the region below this line (including the area through the origin).
By following these steps, the graph of the inequality [tex]\(2x - 3y < 12\)[/tex] will be a shaded area below the line [tex]\(2x - 3y = 12\)[/tex], not including the line itself because the inequality is strictly less than ([tex]\(<\)[/tex]) rather than less than or equal to ([tex]\(\le\)[/tex]).
