Answered

Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

Rewrite the following system of inequalities in a clearer format.

a. \(3x + y \leq 3\)
b. [tex]\(y - 4 \geq -2x\)[/tex]

Sagot :

GinnyAnswer
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]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.