Answer :
To solve the system of inequalities
[tex]\[ \begin{cases} y \leq -7x + 2 \\ y \geq 9x + 7 \\ \end{cases} \][/tex]
by graphing, we need to follow several steps.
### Step 1: Graph the Boundary Lines
1. For [tex]\(y = -7x + 2\)[/tex]:
- This is a straight line. To plot it, we need at least two points.
- When [tex]\(x = 0\)[/tex]:
[tex]\[ y = -7(0) + 2 = 2 \][/tex]
So, one point is [tex]\((0, 2)\)[/tex].
- When [tex]\(x = 1\)[/tex]:
[tex]\[ y = -7(1) + 2 = -5 \][/tex]
So, another point is [tex]\((1, -5)\)[/tex].
- Plot these two points and draw the line [tex]\(y = -7x + 2\)[/tex]. Since the inequality is [tex]\(y \leq -7x + 2\)[/tex], we will shade the region below this line.
2. For [tex]\(y = 9x + 7\)[/tex]:
- This is also a straight line. To plot it, we need at least two points.
- When [tex]\(x = 0\)[/tex]:
[tex]\[ y = 9(0) + 7 = 7 \][/tex]
So, one point is [tex]\((0, 7)\)[/tex].
- When [tex]\(x = 1\)[/tex]:
[tex]\[ y = 9(1) + 7 = 16 \][/tex]
So, another point is [tex]\((1, 16)\)[/tex].
- Plot these two points and draw the line [tex]\(y = 9x + 7\)[/tex]. Since the inequality is [tex]\(y \geq 9x + 7\)[/tex], we will shade the region above this line.
### Step 2: Find the Intersection Point
To find where these lines intersect, set the equations equal to each other:
[tex]\[ -7x + 2 = 9x + 7 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ -7x + 2 = 9x + 7 \implies -7x - 9x = 7 - 2 \implies -16x = 5 \implies x = -\frac{5}{16} \][/tex]
Then, substitute [tex]\(x = -\frac{5}{16}\)[/tex] back into either equation to find [tex]\(y\)[/tex]:
[tex]\[ y = -7 \left(-\frac{5}{16}\right) + 2 = \frac{35}{16} + 2 = \frac{35}{16} + \frac{32}{16} = \frac{67}{16} \][/tex]
So, the intersection point is:
[tex]\[ \left(-\frac{5}{16}, \frac{67}{16}\right) \][/tex]
### Step 3: Shading the Regions
1. The region for [tex]\(y \leq -7x + 2\)[/tex] is below the line [tex]\(y = -7x + 2\)[/tex].
2. The region for [tex]\(y \geq 9x + 7\)[/tex] is above the line [tex]\(y = 9x + 7\)[/tex].
### Step 4: Identify the Intersection Region
Since we are looking for the region that satisfies both inequalities, we identify the area where the shaded regions from [tex]\(y \leq -7x + 2\)[/tex] and [tex]\(y \geq 9x + 7\)[/tex] overlap. However, observing the lines it becomes apparent that the shaded areas do not overlap since one inequality represents a region below a very steep line with a large negative slope, and the other inequality represents a region above a very steep line with a large positive slope. They would intersect only at a single point if extended infinitely, but in practical terms, the regions do not overlap.
### Conclusion
For these sets of inequalities, there is no region in the [tex]\(xy\)[/tex]-plane that satisfies both conditions simultaneously:
[tex]\[ \boxed{\text{No solution}} \][/tex]
[tex]\[ \begin{cases} y \leq -7x + 2 \\ y \geq 9x + 7 \\ \end{cases} \][/tex]
by graphing, we need to follow several steps.
### Step 1: Graph the Boundary Lines
1. For [tex]\(y = -7x + 2\)[/tex]:
- This is a straight line. To plot it, we need at least two points.
- When [tex]\(x = 0\)[/tex]:
[tex]\[ y = -7(0) + 2 = 2 \][/tex]
So, one point is [tex]\((0, 2)\)[/tex].
- When [tex]\(x = 1\)[/tex]:
[tex]\[ y = -7(1) + 2 = -5 \][/tex]
So, another point is [tex]\((1, -5)\)[/tex].
- Plot these two points and draw the line [tex]\(y = -7x + 2\)[/tex]. Since the inequality is [tex]\(y \leq -7x + 2\)[/tex], we will shade the region below this line.
2. For [tex]\(y = 9x + 7\)[/tex]:
- This is also a straight line. To plot it, we need at least two points.
- When [tex]\(x = 0\)[/tex]:
[tex]\[ y = 9(0) + 7 = 7 \][/tex]
So, one point is [tex]\((0, 7)\)[/tex].
- When [tex]\(x = 1\)[/tex]:
[tex]\[ y = 9(1) + 7 = 16 \][/tex]
So, another point is [tex]\((1, 16)\)[/tex].
- Plot these two points and draw the line [tex]\(y = 9x + 7\)[/tex]. Since the inequality is [tex]\(y \geq 9x + 7\)[/tex], we will shade the region above this line.
### Step 2: Find the Intersection Point
To find where these lines intersect, set the equations equal to each other:
[tex]\[ -7x + 2 = 9x + 7 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ -7x + 2 = 9x + 7 \implies -7x - 9x = 7 - 2 \implies -16x = 5 \implies x = -\frac{5}{16} \][/tex]
Then, substitute [tex]\(x = -\frac{5}{16}\)[/tex] back into either equation to find [tex]\(y\)[/tex]:
[tex]\[ y = -7 \left(-\frac{5}{16}\right) + 2 = \frac{35}{16} + 2 = \frac{35}{16} + \frac{32}{16} = \frac{67}{16} \][/tex]
So, the intersection point is:
[tex]\[ \left(-\frac{5}{16}, \frac{67}{16}\right) \][/tex]
### Step 3: Shading the Regions
1. The region for [tex]\(y \leq -7x + 2\)[/tex] is below the line [tex]\(y = -7x + 2\)[/tex].
2. The region for [tex]\(y \geq 9x + 7\)[/tex] is above the line [tex]\(y = 9x + 7\)[/tex].
### Step 4: Identify the Intersection Region
Since we are looking for the region that satisfies both inequalities, we identify the area where the shaded regions from [tex]\(y \leq -7x + 2\)[/tex] and [tex]\(y \geq 9x + 7\)[/tex] overlap. However, observing the lines it becomes apparent that the shaded areas do not overlap since one inequality represents a region below a very steep line with a large negative slope, and the other inequality represents a region above a very steep line with a large positive slope. They would intersect only at a single point if extended infinitely, but in practical terms, the regions do not overlap.
### Conclusion
For these sets of inequalities, there is no region in the [tex]\(xy\)[/tex]-plane that satisfies both conditions simultaneously:
[tex]\[ \boxed{\text{No solution}} \][/tex]
