Sure! Let's go through the given inequalities step-by-step.
### Step 1: Understand and Write the Inequalities
We are given:
1. [tex]\(3x + 2y < 0\)[/tex]
2. [tex]\(y < -\frac{3}{2}x\)[/tex]
### Step 2: Isolate [tex]\( y \)[/tex] in the First Inequality
Let's isolate [tex]\( y \)[/tex] in [tex]\( 3x + 2y < 0 \)[/tex]:
1. Starting with the inequality:
[tex]\[ 3x + 2y < 0 \][/tex]
2. Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 2y < -3x \][/tex]
3. Divide both sides by 2:
[tex]\[ y < -\frac{3}{2}x \][/tex]
So, rewriting the first inequality in terms of [tex]\( y \)[/tex]:
[tex]\[ y < -\frac{3}{2}x \][/tex]
### Step 3: Examine the Second Inequality
The second inequality is already given to us in a form with [tex]\( y \)[/tex] isolated:
[tex]\[ y < -\frac{3}{2}x \][/tex]
### Step 4: Combine the Results
We now compare the two results from our inequalities:
1. From the first inequality:
[tex]\[ y < -\frac{3}{2}x \][/tex]
2. From the second inequality:
[tex]\[ y < -\frac{3}{2}x \][/tex]
Both of these results are the same. This means the regions for [tex]\( y \)[/tex] defined by both inequalities are identical.
Therefore, the solution to the system of inequalities is:
[tex]\[ y < -\frac{3}{2}x \][/tex]
This represents the region in the coordinate plane below the line [tex]\( y = -\frac{3}{2}x \)[/tex].
