Answer :
To graph the linear inequality [tex]\(2x - 3y < 12\)[/tex], you need to follow these steps:
### Step 1: Graph the Boundary Line
1. Rewrite the Inequality as an Equation:
Convert the inequality into an equation to find the boundary line:
[tex]\[ 2x - 3y = 12 \][/tex]
2. Find the Intercepts:
- x-intercept: Set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - 3(0) = 12 \implies 2x = 12 \implies x = 6 \][/tex]
So, the x-intercept is [tex]\( (6, 0) \)[/tex].
- y-intercept: Set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[ 2(0) - 3y = 12 \implies -3y = 12 \implies y = -4 \][/tex]
So, the y-intercept is [tex]\( (0, -4) \)[/tex].
3. Plot the Intercepts:
Mark the points [tex]\( (6, 0) \)[/tex] and [tex]\( (0, -4) \)[/tex] on the coordinate plane.
4. Draw the Boundary Line:
Draw a straight line passing through the points [tex]\( (6, 0) \)[/tex] and [tex]\( (0, -4) \)[/tex]. Since the original inequality is strict ([tex]\(<\)[/tex]), this boundary line should be a dashed line to indicate that points on the line are not included in the solution set.
### Step 2: Determine the Solution Region
1. Test a Point:
Choose a test point not on the boundary line to determine which side of the line satisfies the inequality. The origin [tex]\((0, 0)\)[/tex] is a convenient test point.
- Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality:
[tex]\[ 2(0) - 3(0) < 12 \implies 0 < 12 \][/tex]
Since this is true, the origin [tex]\((0, 0)\)[/tex] is part of the solution set.
2. Shade the Correct Region:
Shade the region of the graph where the inequality [tex]\(2x - 3y < 12\)[/tex] holds true. Since our test point [tex]\((0, 0)\)[/tex] satisfied the inequality, we shade the region that includes the origin and is below the boundary line.
### Graph Summary:
- Draw a dashed line through the points [tex]\( (6, 0) \)[/tex] and [tex]\( (0, -4) \)[/tex].
- Shade the region below this line, which will represent all the solutions to the inequality [tex]\(2x - 3y < 12\)[/tex].
### Visualization:
(Here we describe the final graph, though we cannot literally draw it in this text format.)
- The x-axis intersects at [tex]\(6\)[/tex], and the y-axis intersects at [tex]\(-4\)[/tex].
- A dashed line joins these points.
- The region below the dashed line, extending infinitely to the left and the right, is shaded to indicate the solution to the inequality [tex]\(2x - 3y < 12\)[/tex].
### Step 1: Graph the Boundary Line
1. Rewrite the Inequality as an Equation:
Convert the inequality into an equation to find the boundary line:
[tex]\[ 2x - 3y = 12 \][/tex]
2. Find the Intercepts:
- x-intercept: Set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - 3(0) = 12 \implies 2x = 12 \implies x = 6 \][/tex]
So, the x-intercept is [tex]\( (6, 0) \)[/tex].
- y-intercept: Set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[ 2(0) - 3y = 12 \implies -3y = 12 \implies y = -4 \][/tex]
So, the y-intercept is [tex]\( (0, -4) \)[/tex].
3. Plot the Intercepts:
Mark the points [tex]\( (6, 0) \)[/tex] and [tex]\( (0, -4) \)[/tex] on the coordinate plane.
4. Draw the Boundary Line:
Draw a straight line passing through the points [tex]\( (6, 0) \)[/tex] and [tex]\( (0, -4) \)[/tex]. Since the original inequality is strict ([tex]\(<\)[/tex]), this boundary line should be a dashed line to indicate that points on the line are not included in the solution set.
### Step 2: Determine the Solution Region
1. Test a Point:
Choose a test point not on the boundary line to determine which side of the line satisfies the inequality. The origin [tex]\((0, 0)\)[/tex] is a convenient test point.
- Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality:
[tex]\[ 2(0) - 3(0) < 12 \implies 0 < 12 \][/tex]
Since this is true, the origin [tex]\((0, 0)\)[/tex] is part of the solution set.
2. Shade the Correct Region:
Shade the region of the graph where the inequality [tex]\(2x - 3y < 12\)[/tex] holds true. Since our test point [tex]\((0, 0)\)[/tex] satisfied the inequality, we shade the region that includes the origin and is below the boundary line.
### Graph Summary:
- Draw a dashed line through the points [tex]\( (6, 0) \)[/tex] and [tex]\( (0, -4) \)[/tex].
- Shade the region below this line, which will represent all the solutions to the inequality [tex]\(2x - 3y < 12\)[/tex].
### Visualization:
(Here we describe the final graph, though we cannot literally draw it in this text format.)
- The x-axis intersects at [tex]\(6\)[/tex], and the y-axis intersects at [tex]\(-4\)[/tex].
- A dashed line joins these points.
- The region below the dashed line, extending infinitely to the left and the right, is shaded to indicate the solution to the inequality [tex]\(2x - 3y < 12\)[/tex].
