At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
Sure, let's tackle this system of inequalities step by step. We'll graph both inequalities and determine the solution set.
### Step 1: Graph the first inequality
The first inequality is:
[tex]\[ y \geq \frac{5}{4} x - 8 \][/tex]
To graph this:
1. Start by graphing the boundary line [tex]\( y = \frac{5}{4} x - 8 \)[/tex].
- This line has a slope of [tex]\(\frac{5}{4}\)[/tex] and a y-intercept at [tex]\(y = -8\)[/tex].
- To find more points on the line, you can plug in values for [tex]\(x\)[/tex]:
- When [tex]\(x = 0\)[/tex], [tex]\( y = -8 \)[/tex].
- When [tex]\(x = 4\)[/tex], [tex]\( y = -3 \)[/tex] (using [tex]\( y = \frac{5}{4} (4) - 8 \)[/tex]).
- Plot these points and draw a straight line.
2. Shade the region where [tex]\( y \)[/tex] is greater than or equal to [tex]\( \frac{5}{4} x - 8 \)[/tex].
- Shade above the line since the inequality is [tex]\( \geq \)[/tex].
### Step 2: Graph the second inequality
The second inequality is:
[tex]\[ y < -\frac{1}{2} x - 1 \][/tex]
To graph this:
1. Start by graphing the boundary line [tex]\( y = -\frac{1}{2} x - 1 \)[/tex].
- This line has a slope of [tex]\(-\frac{1}{2}\)[/tex] and a y-intercept at [tex]\(y = -1\)[/tex].
- To find more points on the line, you can plug in values for [tex]\(x\)[/tex]:
- When [tex]\(x = 0\)[/tex], [tex]\( y = -1 \)[/tex].
- When [tex]\(x = 2\)[/tex], [tex]\( y = -2 \)[/tex] (using [tex]\( y = -\frac{1}{2} (2) - 1 \)[/tex]).
- Plot these points and draw a dashed line because the inequality is strict ([tex]\(<\)[/tex], not [tex]\(\leq\)[/tex]).
2. Shade the region where [tex]\( y \)[/tex] is less than [tex]\( -\frac{1}{2} x - 1 \)[/tex].
- Shade below the line since the inequality is [tex]\( < \)[/tex].
### Step 3: Find the intersection point of the boundary lines
To do this, solve the system of equations:
[tex]\[ y = \frac{5}{4} x - 8 \][/tex]
[tex]\[ y = -\frac{1}{2} x - 1 \][/tex]
1. Set the equations equal to each other to find [tex]\(x\)[/tex]:
[tex]\[ \frac{5}{4} x - 8 = -\frac{1}{2} x - 1 \][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{5}{4} x + \frac{1}{2} x = 8 - 1 \][/tex]
Simplify and combine like terms:
[tex]\[ \frac{5}{4} x + \frac{2}{4} x = 7 \][/tex]
[tex]\[ \frac{7}{4} x = 7 \][/tex]
Multiply both sides by [tex]\(\frac{4}{7}\)[/tex]:
[tex]\[ x = 4 \][/tex]
3. Substitute [tex]\(x = 4\)[/tex] back into one of the equations to find [tex]\(y\)[/tex]:
[tex]\[ y = \frac{5}{4} (4) - 8 = 5 - 8 = -3 \][/tex]
Hence, the intersection point is [tex]\((4, -3)\)[/tex].
### Step 4: Determine the solution set
The solution set is the region where the two shaded areas overlap. This is the region above the line [tex]\( y = \frac{5}{4} x - 8 \)[/tex] and below the line [tex]\( y = -\frac{1}{2} x - 1 \)[/tex].
### Step 5: State the coordinates of a point in the solution set
From our intersection point [tex]\((4, -3)\)[/tex], we can verify the region. Any point in the overlapping shaded region will be part of the solution set. For instance, [tex]\((4, -3)\)[/tex] is in the solution set because it satisfies both inequalities.
Therefore, the coordinates of one point in the solution set is:
[tex]\[ (4, -3) \][/tex]
### Step 1: Graph the first inequality
The first inequality is:
[tex]\[ y \geq \frac{5}{4} x - 8 \][/tex]
To graph this:
1. Start by graphing the boundary line [tex]\( y = \frac{5}{4} x - 8 \)[/tex].
- This line has a slope of [tex]\(\frac{5}{4}\)[/tex] and a y-intercept at [tex]\(y = -8\)[/tex].
- To find more points on the line, you can plug in values for [tex]\(x\)[/tex]:
- When [tex]\(x = 0\)[/tex], [tex]\( y = -8 \)[/tex].
- When [tex]\(x = 4\)[/tex], [tex]\( y = -3 \)[/tex] (using [tex]\( y = \frac{5}{4} (4) - 8 \)[/tex]).
- Plot these points and draw a straight line.
2. Shade the region where [tex]\( y \)[/tex] is greater than or equal to [tex]\( \frac{5}{4} x - 8 \)[/tex].
- Shade above the line since the inequality is [tex]\( \geq \)[/tex].
### Step 2: Graph the second inequality
The second inequality is:
[tex]\[ y < -\frac{1}{2} x - 1 \][/tex]
To graph this:
1. Start by graphing the boundary line [tex]\( y = -\frac{1}{2} x - 1 \)[/tex].
- This line has a slope of [tex]\(-\frac{1}{2}\)[/tex] and a y-intercept at [tex]\(y = -1\)[/tex].
- To find more points on the line, you can plug in values for [tex]\(x\)[/tex]:
- When [tex]\(x = 0\)[/tex], [tex]\( y = -1 \)[/tex].
- When [tex]\(x = 2\)[/tex], [tex]\( y = -2 \)[/tex] (using [tex]\( y = -\frac{1}{2} (2) - 1 \)[/tex]).
- Plot these points and draw a dashed line because the inequality is strict ([tex]\(<\)[/tex], not [tex]\(\leq\)[/tex]).
2. Shade the region where [tex]\( y \)[/tex] is less than [tex]\( -\frac{1}{2} x - 1 \)[/tex].
- Shade below the line since the inequality is [tex]\( < \)[/tex].
### Step 3: Find the intersection point of the boundary lines
To do this, solve the system of equations:
[tex]\[ y = \frac{5}{4} x - 8 \][/tex]
[tex]\[ y = -\frac{1}{2} x - 1 \][/tex]
1. Set the equations equal to each other to find [tex]\(x\)[/tex]:
[tex]\[ \frac{5}{4} x - 8 = -\frac{1}{2} x - 1 \][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{5}{4} x + \frac{1}{2} x = 8 - 1 \][/tex]
Simplify and combine like terms:
[tex]\[ \frac{5}{4} x + \frac{2}{4} x = 7 \][/tex]
[tex]\[ \frac{7}{4} x = 7 \][/tex]
Multiply both sides by [tex]\(\frac{4}{7}\)[/tex]:
[tex]\[ x = 4 \][/tex]
3. Substitute [tex]\(x = 4\)[/tex] back into one of the equations to find [tex]\(y\)[/tex]:
[tex]\[ y = \frac{5}{4} (4) - 8 = 5 - 8 = -3 \][/tex]
Hence, the intersection point is [tex]\((4, -3)\)[/tex].
### Step 4: Determine the solution set
The solution set is the region where the two shaded areas overlap. This is the region above the line [tex]\( y = \frac{5}{4} x - 8 \)[/tex] and below the line [tex]\( y = -\frac{1}{2} x - 1 \)[/tex].
### Step 5: State the coordinates of a point in the solution set
From our intersection point [tex]\((4, -3)\)[/tex], we can verify the region. Any point in the overlapping shaded region will be part of the solution set. For instance, [tex]\((4, -3)\)[/tex] is in the solution set because it satisfies both inequalities.
Therefore, the coordinates of one point in the solution set is:
[tex]\[ (4, -3) \][/tex]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
