Answer :
To solve the system of inequalities:
[tex]\[ 3x + y \leq 3 \][/tex]
[tex]\[ y - 4 \geq -2x \][/tex]
we will first simplify and rewrite them in more recognizable forms, then solve and analyze them step-by-step.
### Step 1: Simplify the second inequality.
The second inequality is:
[tex]\[ y - 4 \geq -2x \][/tex]
To isolate \( y \), we add 4 to both sides:
[tex]\[ y \geq -2x + 4 \][/tex]
Now, the inequalities are:
[tex]\[ 3x + y \leq 3 \][/tex]
[tex]\[ y \geq -2x + 4 \][/tex]
### Step 2: Interpret the inequalities.
- The first inequality, \( 3x + y \leq 3 \), can be interpreted as the region on or below the line \( y = -3x + 3 \).
- The second inequality, \( y \geq -2x + 4 \), can be interpreted as the region on or above the line \( y = -2x + 4 \).
### Step 3: Graph the inequalities.
On a coordinate plane:
1. For \( 3x + y \leq 3 \) or \( y \leq -3x + 3 \):
- The boundary line is \( y = -3x + 3 \). It is a straight line with a slope of -3 and y-intercept at 3.
- Shade the region below this line including the line itself since it includes the "=" part.
2. For \( y \geq -2x + 4 \):
- The boundary line is \( y = -2x + 4 \). It is a straight line with a slope of -2 and y-intercept at 4.
- Shade the region above this line including the line itself since it includes the "=" part.
### Step 4: Find the solution set.
The solution set is the region where the shaded areas overlap. This is the part of the coordinate plane that satisfies both inequalities simultaneously.
### Step 5: Check intersections.
To better understand the solution set, we can find where the boundary lines intersect:
Set \( -3x + 3 = -2x + 4 \) and solve for \( x \):
[tex]\[ -3x + 3 = -2x + 4 \][/tex]
[tex]\[ -x = 1 \][/tex]
[tex]\[ x = -1 \][/tex]
Plug \( x = -1 \) back into either equation to find \( y \):
[tex]\[ y = -3(-1) + 3 = 3 + 3 = 6 \][/tex]
So, the lines intersect at the point \((-1, 6)\).
### Conclusion:
The solution to the system of inequalities is the region bounded by the lines \( y = -3x + 3 \) and \( y = -2x + 4 \). This region includes the intersection point \((-1, 6)\) and extends infinitely within the defined shaded areas.
Therefore, the inequalities in their simplified forms are:
[tex]\[ 3x + y \leq 3 \][/tex]
[tex]\[ y \geq -2x + 4 \][/tex]
[tex]\[ 3x + y \leq 3 \][/tex]
[tex]\[ y - 4 \geq -2x \][/tex]
we will first simplify and rewrite them in more recognizable forms, then solve and analyze them step-by-step.
### Step 1: Simplify the second inequality.
The second inequality is:
[tex]\[ y - 4 \geq -2x \][/tex]
To isolate \( y \), we add 4 to both sides:
[tex]\[ y \geq -2x + 4 \][/tex]
Now, the inequalities are:
[tex]\[ 3x + y \leq 3 \][/tex]
[tex]\[ y \geq -2x + 4 \][/tex]
### Step 2: Interpret the inequalities.
- The first inequality, \( 3x + y \leq 3 \), can be interpreted as the region on or below the line \( y = -3x + 3 \).
- The second inequality, \( y \geq -2x + 4 \), can be interpreted as the region on or above the line \( y = -2x + 4 \).
### Step 3: Graph the inequalities.
On a coordinate plane:
1. For \( 3x + y \leq 3 \) or \( y \leq -3x + 3 \):
- The boundary line is \( y = -3x + 3 \). It is a straight line with a slope of -3 and y-intercept at 3.
- Shade the region below this line including the line itself since it includes the "=" part.
2. For \( y \geq -2x + 4 \):
- The boundary line is \( y = -2x + 4 \). It is a straight line with a slope of -2 and y-intercept at 4.
- Shade the region above this line including the line itself since it includes the "=" part.
### Step 4: Find the solution set.
The solution set is the region where the shaded areas overlap. This is the part of the coordinate plane that satisfies both inequalities simultaneously.
### Step 5: Check intersections.
To better understand the solution set, we can find where the boundary lines intersect:
Set \( -3x + 3 = -2x + 4 \) and solve for \( x \):
[tex]\[ -3x + 3 = -2x + 4 \][/tex]
[tex]\[ -x = 1 \][/tex]
[tex]\[ x = -1 \][/tex]
Plug \( x = -1 \) back into either equation to find \( y \):
[tex]\[ y = -3(-1) + 3 = 3 + 3 = 6 \][/tex]
So, the lines intersect at the point \((-1, 6)\).
### Conclusion:
The solution to the system of inequalities is the region bounded by the lines \( y = -3x + 3 \) and \( y = -2x + 4 \). This region includes the intersection point \((-1, 6)\) and extends infinitely within the defined shaded areas.
Therefore, the inequalities in their simplified forms are:
[tex]\[ 3x + y \leq 3 \][/tex]
[tex]\[ y \geq -2x + 4 \][/tex]
