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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]
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