At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To solve the system of inequalities:
[tex]\[ \begin{cases} x + 4y \geq 10 \\ 3x - 2y < 12 \end{cases} \][/tex]
we need to determine the region in the [tex]\(xy\)[/tex]-plane that satisfies both inequalities. We'll proceed step-by-step by analyzing each inequality and then finding the intersection of the regions they define.
### Analyzing the First Inequality: [tex]\(x + 4y \geq 10\)[/tex]
1. Rewrite in Slope-Intercept Form:
[tex]\[ x + 4y = 10 \quad \Rightarrow \quad 4y = -x + 10 \quad \Rightarrow \quad y = -\frac{1}{4}x + \frac{10}{4} \quad \Rightarrow \quad y = -\frac{1}{4}x + 2.5 \][/tex]
2. Graph the Equality:
- This line has a y-intercept of [tex]\(2.5\)[/tex] and a slope of [tex]\(-\frac{1}{4}\)[/tex].
- The line [tex]\(y = -\frac{1}{4}x + 2.5\)[/tex] divides the plane.
3. Determine the Region:
- To figure out which side of the line is valid, pick a test point such as (0, 0).
- Substitute into the inequality: [tex]\(0 + 4(0) = 0\)[/tex], we see that [tex]\(0 \geq 10\)[/tex] is false.
- So, the region that satisfies [tex]\(x + 4y \geq 10\)[/tex] is the region above the line [tex]\(y = -\frac{1}{4}x + 2.5\)[/tex].
### Analyzing the Second Inequality: [tex]\(3x - 2y < 12\)[/tex]
1. Rewrite in Slope-Intercept Form:
[tex]\[ 3x - 2y = 12 \quad \Rightarrow \quad -2y = -3x + 12 \quad \Rightarrow \quad y = \frac{3}{2}x - 6 \][/tex]
2. Graph the Equality:
- This line has a y-intercept of [tex]\(-6\)[/tex] and a slope of [tex]\(\frac{3}{2}\)[/tex].
- The line [tex]\(y = \frac{3}{2}x - 6\)[/tex] divides the plane.
3. Determine the Region:
- To figure out which side of the line is valid, pick a test point such as [tex]\((0, 0)\)[/tex].
- Substitute into the inequality: [tex]\(3(0) - 2(0) = 0\)[/tex], we see that [tex]\(0 < 12\)[/tex] is true.
- So, the region that satisfies [tex]\(3x - 2y < 12\)[/tex] is the region below the line [tex]\(y = \frac{3}{2}x - 6\)[/tex].
### Finding the Intersection of the Regions
We now seek the overlapping region that satisfies both inequalities:
1. Graphical Solution:
- Draw the lines [tex]\(y = -\frac{1}{4}x + 2.5\)[/tex] and [tex]\(y = \frac{3}{2}x - 6\)[/tex] on the same graph.
- Identify the region that is both above the line [tex]\(y = -\frac{1}{4}x + 2.5\)[/tex] and below the line [tex]\(y = \frac{3}{2}x - 6\)[/tex].
2. Points of Intersection:
- Find where the lines intersect by solving the system:
[tex]\[ -\frac{1}{4}x + 2.5 = \frac{3}{2}x - 6 \][/tex]
Multiply through by 4 to clear fractions:
[tex]\[ -x + 10 = 6x - 24 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ 10 + 24 = 7x \quad \Rightarrow \quad 34 = 7x \quad \Rightarrow \quad x = \frac{34}{7} \approx 4.857 \][/tex]
Substitute [tex]\(x = \frac{34}{7}\)[/tex] back into either line equation to find [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{1}{4}\left(\frac{34}{7}\right) + 2.5 = \frac{-34}{28} + 2.5 = -\frac{17}{14} + 2.5 = -\frac{17}{14} + \frac{35}{14} = \frac{18}{14} = \frac{9}{7} \approx 1.286 \][/tex]
Therefore, the lines intersect at approximately [tex]\(\left(\frac{34}{7}, \frac{9}{7}\right)\)[/tex].
### Conclusion
The solution to the system of inequalities is the region in the [tex]\(xy\)[/tex]-plane that:
- Lies above the line [tex]\(y = -\frac{1}{4}x + 2.5\)[/tex]
- Lies below the line [tex]\(y = \frac{3}{2}x - 6\)[/tex]
Graphing these regions and finding their overlap will visually provide the area satisfying both conditions.
[tex]\[ \begin{cases} x + 4y \geq 10 \\ 3x - 2y < 12 \end{cases} \][/tex]
we need to determine the region in the [tex]\(xy\)[/tex]-plane that satisfies both inequalities. We'll proceed step-by-step by analyzing each inequality and then finding the intersection of the regions they define.
### Analyzing the First Inequality: [tex]\(x + 4y \geq 10\)[/tex]
1. Rewrite in Slope-Intercept Form:
[tex]\[ x + 4y = 10 \quad \Rightarrow \quad 4y = -x + 10 \quad \Rightarrow \quad y = -\frac{1}{4}x + \frac{10}{4} \quad \Rightarrow \quad y = -\frac{1}{4}x + 2.5 \][/tex]
2. Graph the Equality:
- This line has a y-intercept of [tex]\(2.5\)[/tex] and a slope of [tex]\(-\frac{1}{4}\)[/tex].
- The line [tex]\(y = -\frac{1}{4}x + 2.5\)[/tex] divides the plane.
3. Determine the Region:
- To figure out which side of the line is valid, pick a test point such as (0, 0).
- Substitute into the inequality: [tex]\(0 + 4(0) = 0\)[/tex], we see that [tex]\(0 \geq 10\)[/tex] is false.
- So, the region that satisfies [tex]\(x + 4y \geq 10\)[/tex] is the region above the line [tex]\(y = -\frac{1}{4}x + 2.5\)[/tex].
### Analyzing the Second Inequality: [tex]\(3x - 2y < 12\)[/tex]
1. Rewrite in Slope-Intercept Form:
[tex]\[ 3x - 2y = 12 \quad \Rightarrow \quad -2y = -3x + 12 \quad \Rightarrow \quad y = \frac{3}{2}x - 6 \][/tex]
2. Graph the Equality:
- This line has a y-intercept of [tex]\(-6\)[/tex] and a slope of [tex]\(\frac{3}{2}\)[/tex].
- The line [tex]\(y = \frac{3}{2}x - 6\)[/tex] divides the plane.
3. Determine the Region:
- To figure out which side of the line is valid, pick a test point such as [tex]\((0, 0)\)[/tex].
- Substitute into the inequality: [tex]\(3(0) - 2(0) = 0\)[/tex], we see that [tex]\(0 < 12\)[/tex] is true.
- So, the region that satisfies [tex]\(3x - 2y < 12\)[/tex] is the region below the line [tex]\(y = \frac{3}{2}x - 6\)[/tex].
### Finding the Intersection of the Regions
We now seek the overlapping region that satisfies both inequalities:
1. Graphical Solution:
- Draw the lines [tex]\(y = -\frac{1}{4}x + 2.5\)[/tex] and [tex]\(y = \frac{3}{2}x - 6\)[/tex] on the same graph.
- Identify the region that is both above the line [tex]\(y = -\frac{1}{4}x + 2.5\)[/tex] and below the line [tex]\(y = \frac{3}{2}x - 6\)[/tex].
2. Points of Intersection:
- Find where the lines intersect by solving the system:
[tex]\[ -\frac{1}{4}x + 2.5 = \frac{3}{2}x - 6 \][/tex]
Multiply through by 4 to clear fractions:
[tex]\[ -x + 10 = 6x - 24 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ 10 + 24 = 7x \quad \Rightarrow \quad 34 = 7x \quad \Rightarrow \quad x = \frac{34}{7} \approx 4.857 \][/tex]
Substitute [tex]\(x = \frac{34}{7}\)[/tex] back into either line equation to find [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{1}{4}\left(\frac{34}{7}\right) + 2.5 = \frac{-34}{28} + 2.5 = -\frac{17}{14} + 2.5 = -\frac{17}{14} + \frac{35}{14} = \frac{18}{14} = \frac{9}{7} \approx 1.286 \][/tex]
Therefore, the lines intersect at approximately [tex]\(\left(\frac{34}{7}, \frac{9}{7}\right)\)[/tex].
### Conclusion
The solution to the system of inequalities is the region in the [tex]\(xy\)[/tex]-plane that:
- Lies above the line [tex]\(y = -\frac{1}{4}x + 2.5\)[/tex]
- Lies below the line [tex]\(y = \frac{3}{2}x - 6\)[/tex]
Graphing these regions and finding their overlap will visually provide the area satisfying both conditions.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.