Answer :
To solve the system of inequalities:
1. [tex]\(3x + 2y < 0\)[/tex]
2. [tex]\(y < -\frac{3}{2}x\)[/tex]
we need to find the region on the coordinate plane that satisfies both conditions.
### Step-by-Step Solution:
#### Inequality 1: [tex]\(3x + 2y < 0\)[/tex]
1. Consider the equation [tex]\(3x + 2y = 0\)[/tex] as a boundary.
2. Solve for [tex]\(y\)[/tex]:
[tex]\[ 2y = -3x \implies y = -\frac{3}{2}x \][/tex]
3. Plot the line [tex]\(y = -\frac{3}{2}x\)[/tex] on the coordinate plane. This line will divide the plane into two regions.
4. Since the inequality is [tex]\(3x + 2y < 0\)[/tex], the region of interest is below the line [tex]\(y = -\frac{3}{2}x\)[/tex], because [tex]\(y\)[/tex] must be less than [tex]\(-\frac{3}{2}x\)[/tex].
#### Inequality 2: [tex]\(y < -\frac{3}{2}x\)[/tex]
1. This inequality describes the same boundary line as in inequality 1: [tex]\(y = -\frac{3}{2}x\)[/tex].
2. According to this inequality, the region of interest is also below the line [tex]\(y = -\frac{3}{2}x\)[/tex].
### Combining the Inequalities
Since both inequalities require the region to be below the line [tex]\(y = -\frac{3}{2}x\)[/tex], the solution to the system of inequalities is the region below this line.
To summarize:
- The line [tex]\(y = -\frac{3}{2}x\)[/tex] acts as a boundary.
- The solution region for both inequalities is the area below the line [tex]\(y = -\frac{3}{2}x\)[/tex].
Hence, the solution for the system of inequalities is the set of all points [tex]\((x, y)\)[/tex] that lie below the line [tex]\(y = -\frac{3}{2}x\)[/tex]. This means:
[tex]\[ 3x + 2y < 0 \quad \text{and} \quad y < -\frac{3}{2}x. \][/tex]
1. [tex]\(3x + 2y < 0\)[/tex]
2. [tex]\(y < -\frac{3}{2}x\)[/tex]
we need to find the region on the coordinate plane that satisfies both conditions.
### Step-by-Step Solution:
#### Inequality 1: [tex]\(3x + 2y < 0\)[/tex]
1. Consider the equation [tex]\(3x + 2y = 0\)[/tex] as a boundary.
2. Solve for [tex]\(y\)[/tex]:
[tex]\[ 2y = -3x \implies y = -\frac{3}{2}x \][/tex]
3. Plot the line [tex]\(y = -\frac{3}{2}x\)[/tex] on the coordinate plane. This line will divide the plane into two regions.
4. Since the inequality is [tex]\(3x + 2y < 0\)[/tex], the region of interest is below the line [tex]\(y = -\frac{3}{2}x\)[/tex], because [tex]\(y\)[/tex] must be less than [tex]\(-\frac{3}{2}x\)[/tex].
#### Inequality 2: [tex]\(y < -\frac{3}{2}x\)[/tex]
1. This inequality describes the same boundary line as in inequality 1: [tex]\(y = -\frac{3}{2}x\)[/tex].
2. According to this inequality, the region of interest is also below the line [tex]\(y = -\frac{3}{2}x\)[/tex].
### Combining the Inequalities
Since both inequalities require the region to be below the line [tex]\(y = -\frac{3}{2}x\)[/tex], the solution to the system of inequalities is the region below this line.
To summarize:
- The line [tex]\(y = -\frac{3}{2}x\)[/tex] acts as a boundary.
- The solution region for both inequalities is the area below the line [tex]\(y = -\frac{3}{2}x\)[/tex].
Hence, the solution for the system of inequalities is the set of all points [tex]\((x, y)\)[/tex] that lie below the line [tex]\(y = -\frac{3}{2}x\)[/tex]. This means:
[tex]\[ 3x + 2y < 0 \quad \text{and} \quad y < -\frac{3}{2}x. \][/tex]
