Answer :
To solve and graph the system of inequalities, follow these steps:
### Step 1: Rewrite the inequalities in slope-intercept form (i.e., [tex]\( y = mx + b \)[/tex]).
1. First Inequality: [tex]\( 3y > 2x + 12 \)[/tex]
Divide all terms by 3 to isolate [tex]\( y \)[/tex]:
[tex]\[ y > \frac{2}{3}x + 4 \][/tex]
2. Second Inequality: [tex]\( 2x + y \leq -5 \)[/tex]
Subtract [tex]\( 2x \)[/tex] from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y \leq -2x - 5 \][/tex]
### Step 2: Graph the boundary lines for each inequality as if they were equalities.
1. Boundary Line for [tex]\( y = \frac{2}{3}x + 4 \)[/tex]
- This line has a slope ([tex]\( m \)[/tex]) of [tex]\( \frac{2}{3} \)[/tex] and a y-intercept ([tex]\( b \)[/tex]) of 4.
- Plot the y-intercept at (0, 4).
- Use the slope to find another point: from (0, 4), move up 2 units and right 3 units to (3, 6).
- Draw a dashed line through these points because the inequality is [tex]\( > \)[/tex], not [tex]\( \geq \)[/tex].
2. Boundary Line for [tex]\( y = -2x - 5 \)[/tex]
- This line has a slope ([tex]\( m \)[/tex]) of [tex]\( -2 \)[/tex] and a y-intercept ([tex]\( b \)[/tex]) of -5.
- Plot the y-intercept at (0, -5).
- Use the slope to find another point: from (0, -5), move down 2 units and right 1 unit to (1, -7).
- Draw a solid line through these points because the inequality is [tex]\( \leq \)[/tex], which includes the boundary.
### Step 3: Determine which side of each boundary line represents the solution to the inequality.
1. For [tex]\( y > \frac{2}{3}x + 4 \)[/tex]
- Choose a test point not on the line, such as (0, 0).
- Substitute into the inequality: [tex]\( 0 \stackrel{?}{>} \frac{2}{3}(0) + 4 \rightarrow 0 \stackrel{?}{>} 4 \)[/tex].
- This is false, so the shading is above the line where the inequality holds.
2. For [tex]\( y \leq -2x - 5 \)[/tex]
- Choose a test point not on the line, such as (0, 0).
- Substitute into the inequality: [tex]\( 0 \stackrel{?}{\leq} -2(0) - 5 \rightarrow 0 \stackrel{?}{\leq} -5 \)[/tex].
- This is false, so the shading is below the line where the inequality holds.
### Step 4: Identify the overlapping region.
- The region that satisfies both inequalities is where the shaded areas overlap.
- For clarity:
- Shade above the dashed line [tex]\( y = \frac{2}{3}x + 4 \)[/tex].
- Shade below the solid line [tex]\( y = -2x - 5 \)[/tex].
### Step 5: Graph the solution on the coordinate plane.
1. Dashed Line ( [tex]\( y = \frac{2}{3}x + 4 \)[/tex] )
- Starts at (0, 4), passes through (3, 6), and extends in both directions with a dashed pattern.
- Shade above this line.
2. Solid Line ( [tex]\( y = -2x - 5 \)[/tex] )
- Starts at (0, -5), passes through (1, -7), and extends in both directions with a solid pattern.
- Shade below this line.
3. Overlapping region
- The area where the shadings from both inequalities overlap is the solution region.
Here's a graph illustrating the solution:
```plaintext
y
|
15 |
10 |
5 | _____________
0 |__________0__________
-5 |_________ __________ ____
-10 |
-15 |
-10 -5 0 5 10 x
```
(Note: Graphing tools cannot be rendered here directly, so you need to manually draw this on graph paper or using a graphing tool.)
### Step 1: Rewrite the inequalities in slope-intercept form (i.e., [tex]\( y = mx + b \)[/tex]).
1. First Inequality: [tex]\( 3y > 2x + 12 \)[/tex]
Divide all terms by 3 to isolate [tex]\( y \)[/tex]:
[tex]\[ y > \frac{2}{3}x + 4 \][/tex]
2. Second Inequality: [tex]\( 2x + y \leq -5 \)[/tex]
Subtract [tex]\( 2x \)[/tex] from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y \leq -2x - 5 \][/tex]
### Step 2: Graph the boundary lines for each inequality as if they were equalities.
1. Boundary Line for [tex]\( y = \frac{2}{3}x + 4 \)[/tex]
- This line has a slope ([tex]\( m \)[/tex]) of [tex]\( \frac{2}{3} \)[/tex] and a y-intercept ([tex]\( b \)[/tex]) of 4.
- Plot the y-intercept at (0, 4).
- Use the slope to find another point: from (0, 4), move up 2 units and right 3 units to (3, 6).
- Draw a dashed line through these points because the inequality is [tex]\( > \)[/tex], not [tex]\( \geq \)[/tex].
2. Boundary Line for [tex]\( y = -2x - 5 \)[/tex]
- This line has a slope ([tex]\( m \)[/tex]) of [tex]\( -2 \)[/tex] and a y-intercept ([tex]\( b \)[/tex]) of -5.
- Plot the y-intercept at (0, -5).
- Use the slope to find another point: from (0, -5), move down 2 units and right 1 unit to (1, -7).
- Draw a solid line through these points because the inequality is [tex]\( \leq \)[/tex], which includes the boundary.
### Step 3: Determine which side of each boundary line represents the solution to the inequality.
1. For [tex]\( y > \frac{2}{3}x + 4 \)[/tex]
- Choose a test point not on the line, such as (0, 0).
- Substitute into the inequality: [tex]\( 0 \stackrel{?}{>} \frac{2}{3}(0) + 4 \rightarrow 0 \stackrel{?}{>} 4 \)[/tex].
- This is false, so the shading is above the line where the inequality holds.
2. For [tex]\( y \leq -2x - 5 \)[/tex]
- Choose a test point not on the line, such as (0, 0).
- Substitute into the inequality: [tex]\( 0 \stackrel{?}{\leq} -2(0) - 5 \rightarrow 0 \stackrel{?}{\leq} -5 \)[/tex].
- This is false, so the shading is below the line where the inequality holds.
### Step 4: Identify the overlapping region.
- The region that satisfies both inequalities is where the shaded areas overlap.
- For clarity:
- Shade above the dashed line [tex]\( y = \frac{2}{3}x + 4 \)[/tex].
- Shade below the solid line [tex]\( y = -2x - 5 \)[/tex].
### Step 5: Graph the solution on the coordinate plane.
1. Dashed Line ( [tex]\( y = \frac{2}{3}x + 4 \)[/tex] )
- Starts at (0, 4), passes through (3, 6), and extends in both directions with a dashed pattern.
- Shade above this line.
2. Solid Line ( [tex]\( y = -2x - 5 \)[/tex] )
- Starts at (0, -5), passes through (1, -7), and extends in both directions with a solid pattern.
- Shade below this line.
3. Overlapping region
- The area where the shadings from both inequalities overlap is the solution region.
Here's a graph illustrating the solution:
```plaintext
y
|
15 |
10 |
5 | _____________
0 |__________0__________
-5 |_________ __________ ____
-10 |
-15 |
-10 -5 0 5 10 x
```
(Note: Graphing tools cannot be rendered here directly, so you need to manually draw this on graph paper or using a graphing tool.)
