Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To solve this system of inequalities, we need to analyze and graph the two lines represented by:
1. [tex]\( y = -5x + 2 \)[/tex]
2. [tex]\( y = 3x - 1.5 \)[/tex]
### Step-by-Step Solution
1. Find the intersection point of the two lines:
To find where the two lines intersect, we set the equations equal to each other:
[tex]\[ -5x + 2 = 3x - 1.5 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -5x - 3x + 2 = -1.5 \\ -8x + 2 = -1.5 \\ -8x = -1.5 - 2 \\ -8x = -3.5 \\ x = \frac{-3.5}{-8} \\ x = 0.4375 \][/tex]
Now, solve for [tex]\( y \)[/tex] using [tex]\( x = 0.4375 \)[/tex] in either of the original equations. Using [tex]\( y = -5x + 2 \)[/tex]:
[tex]\[ y = -5(0.4375) + 2 \\ y = -2.1875 + 2 \\ y = -0.1875 \][/tex]
The intersection point is [tex]\( (0.4375, -0.1875) \)[/tex].
2. Graph the lines:
- The line [tex]\( y = -5x + 2 \)[/tex]: This line has a slope of -5 and a y-intercept at [tex]\( (0, 2) \)[/tex].
- The line [tex]\( y = 3x - 1.5 \)[/tex]: This line has a slope of 3 and a y-intercept at [tex]\( (0, -1.5) \)[/tex].
3. Shading the regions:
- For the inequality [tex]\( y \geq -5x + 2 \)[/tex], shade the region above or on the line [tex]\( y = -5x + 2 \)[/tex].
- For the inequality [tex]\( y > 3x - 1.5 \)[/tex], shade the region strictly above the line [tex]\( y = 3x - 1.5 \)[/tex] (note that this does not include the line itself).
4. Determine the feasible region:
- The feasible region is the area that satisfies both inequalities simultaneously. This is the region above the line [tex]\( y = -5x + 2 \)[/tex] AND strictly above [tex]\( y = 3x - 1.5 \)[/tex].
### Final Graph
The graph should include:
- The line [tex]\( y = -5x + 2 \)[/tex] in solid, indicating the boundary for the [tex]\(\geq\)[/tex] inequality, with shading above the line including the line itself.
- The line [tex]\( y = 3x - 1.5 \)[/tex] in dashed, indicating the boundary for the [tex]\(>\)[/tex] inequality, with shading strictly above the line excluding the line itself.
- The intersection point at [tex]\( (0.4375, -0.1875) \)[/tex].
Look for the graph that illustrates the above description.
1. [tex]\( y = -5x + 2 \)[/tex]
2. [tex]\( y = 3x - 1.5 \)[/tex]
### Step-by-Step Solution
1. Find the intersection point of the two lines:
To find where the two lines intersect, we set the equations equal to each other:
[tex]\[ -5x + 2 = 3x - 1.5 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -5x - 3x + 2 = -1.5 \\ -8x + 2 = -1.5 \\ -8x = -1.5 - 2 \\ -8x = -3.5 \\ x = \frac{-3.5}{-8} \\ x = 0.4375 \][/tex]
Now, solve for [tex]\( y \)[/tex] using [tex]\( x = 0.4375 \)[/tex] in either of the original equations. Using [tex]\( y = -5x + 2 \)[/tex]:
[tex]\[ y = -5(0.4375) + 2 \\ y = -2.1875 + 2 \\ y = -0.1875 \][/tex]
The intersection point is [tex]\( (0.4375, -0.1875) \)[/tex].
2. Graph the lines:
- The line [tex]\( y = -5x + 2 \)[/tex]: This line has a slope of -5 and a y-intercept at [tex]\( (0, 2) \)[/tex].
- The line [tex]\( y = 3x - 1.5 \)[/tex]: This line has a slope of 3 and a y-intercept at [tex]\( (0, -1.5) \)[/tex].
3. Shading the regions:
- For the inequality [tex]\( y \geq -5x + 2 \)[/tex], shade the region above or on the line [tex]\( y = -5x + 2 \)[/tex].
- For the inequality [tex]\( y > 3x - 1.5 \)[/tex], shade the region strictly above the line [tex]\( y = 3x - 1.5 \)[/tex] (note that this does not include the line itself).
4. Determine the feasible region:
- The feasible region is the area that satisfies both inequalities simultaneously. This is the region above the line [tex]\( y = -5x + 2 \)[/tex] AND strictly above [tex]\( y = 3x - 1.5 \)[/tex].
### Final Graph
The graph should include:
- The line [tex]\( y = -5x + 2 \)[/tex] in solid, indicating the boundary for the [tex]\(\geq\)[/tex] inequality, with shading above the line including the line itself.
- The line [tex]\( y = 3x - 1.5 \)[/tex] in dashed, indicating the boundary for the [tex]\(>\)[/tex] inequality, with shading strictly above the line excluding the line itself.
- The intersection point at [tex]\( (0.4375, -0.1875) \)[/tex].
Look for the graph that illustrates the above description.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
