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Sagot :
Let's graph the solution to the system of inequalities:
[tex]\[ \begin{array}{l} y \leq -2x - 5 \\ y > 4x - 7 \end{array} \][/tex]
### Step-by-Step Solution:
1. Graph the boundary lines:
- For the inequality [tex]\(y \leq -2x - 5\)[/tex], the boundary line is [tex]\(y = -2x - 5\)[/tex].
- For the inequality [tex]\(y > 4x - 7\)[/tex], the boundary line is [tex]\(y = 4x - 7\)[/tex].
2. Determine the intersection point of the boundary lines:
To find the intersection point, solve the system of equations:
[tex]\[ \begin{cases} y = -2x - 5 \\ y = 4x - 7 \end{cases} \][/tex]
- Equate the right-hand sides of both equations to find [tex]\(x\)[/tex]:
[tex]\[ -2x - 5 = 4x - 7 \][/tex]
[tex]\[ -2x - 4x = -7 + 5 \][/tex]
[tex]\[ -6x = -2 \][/tex]
[tex]\[ x = \frac{1}{3} \][/tex]
- Substitute [tex]\(x = \frac{1}{3}\)[/tex] into one of the equations to find [tex]\(y\)[/tex]:
[tex]\[ y = 4\left(\frac{1}{3}\right) - 7 = \frac{4}{3} - 7 = \frac{4}{3} - \frac{21}{3} = -\frac{17}{3} \][/tex]
Thus, the intersection point is [tex]\(\left(\frac{1}{3}, -\frac{17}{3}\right)\)[/tex].
3. Graph the inequalities:
- For [tex]\(y = -2x - 5\)[/tex]:
- Plot the boundary line [tex]\(y = -2x - 5\)[/tex]. This is a straight line with a slope of [tex]\(-2\)[/tex] and a y-intercept of [tex]\(-5\)[/tex].
- Since the inequality is [tex]\(y \leq -2x - 5\)[/tex], shade the region below the line, including the line itself.
- For [tex]\(y = 4x - 7\)[/tex]:
- Plot the boundary line [tex]\(y = 4x - 7\)[/tex]. This is a straight line with a slope of [tex]\(4\)[/tex] and a y-intercept of [tex]\(-7\)[/tex].
- Since the inequality is [tex]\(y > 4x - 7\)[/tex], shade the region above the line. The line itself is not included (dashed line).
4. Combine the shaded regions:
- The solution to the system is where the shaded regions overlap.
- Identify the overlapping region visually on the graph. This is the region where [tex]\(y \leq -2x - 5\)[/tex] and [tex]\(y > 4x - 7\)[/tex] are both satisfied.
### Conclusion:
To graphically represent the solution, place the two lines on the coordinate plane, shade the appropriate regions based on the inequalities, and identify the intersection point [tex]\(\left(\frac{1}{3}, -\frac{17}{3}\right)\)[/tex]. The feasible region is the overlapping shaded area, which represents the solution to the system of inequalities.
[tex]\[ \begin{array}{l} y \leq -2x - 5 \\ y > 4x - 7 \end{array} \][/tex]
### Step-by-Step Solution:
1. Graph the boundary lines:
- For the inequality [tex]\(y \leq -2x - 5\)[/tex], the boundary line is [tex]\(y = -2x - 5\)[/tex].
- For the inequality [tex]\(y > 4x - 7\)[/tex], the boundary line is [tex]\(y = 4x - 7\)[/tex].
2. Determine the intersection point of the boundary lines:
To find the intersection point, solve the system of equations:
[tex]\[ \begin{cases} y = -2x - 5 \\ y = 4x - 7 \end{cases} \][/tex]
- Equate the right-hand sides of both equations to find [tex]\(x\)[/tex]:
[tex]\[ -2x - 5 = 4x - 7 \][/tex]
[tex]\[ -2x - 4x = -7 + 5 \][/tex]
[tex]\[ -6x = -2 \][/tex]
[tex]\[ x = \frac{1}{3} \][/tex]
- Substitute [tex]\(x = \frac{1}{3}\)[/tex] into one of the equations to find [tex]\(y\)[/tex]:
[tex]\[ y = 4\left(\frac{1}{3}\right) - 7 = \frac{4}{3} - 7 = \frac{4}{3} - \frac{21}{3} = -\frac{17}{3} \][/tex]
Thus, the intersection point is [tex]\(\left(\frac{1}{3}, -\frac{17}{3}\right)\)[/tex].
3. Graph the inequalities:
- For [tex]\(y = -2x - 5\)[/tex]:
- Plot the boundary line [tex]\(y = -2x - 5\)[/tex]. This is a straight line with a slope of [tex]\(-2\)[/tex] and a y-intercept of [tex]\(-5\)[/tex].
- Since the inequality is [tex]\(y \leq -2x - 5\)[/tex], shade the region below the line, including the line itself.
- For [tex]\(y = 4x - 7\)[/tex]:
- Plot the boundary line [tex]\(y = 4x - 7\)[/tex]. This is a straight line with a slope of [tex]\(4\)[/tex] and a y-intercept of [tex]\(-7\)[/tex].
- Since the inequality is [tex]\(y > 4x - 7\)[/tex], shade the region above the line. The line itself is not included (dashed line).
4. Combine the shaded regions:
- The solution to the system is where the shaded regions overlap.
- Identify the overlapping region visually on the graph. This is the region where [tex]\(y \leq -2x - 5\)[/tex] and [tex]\(y > 4x - 7\)[/tex] are both satisfied.
### Conclusion:
To graphically represent the solution, place the two lines on the coordinate plane, shade the appropriate regions based on the inequalities, and identify the intersection point [tex]\(\left(\frac{1}{3}, -\frac{17}{3}\right)\)[/tex]. The feasible region is the overlapping shaded area, which represents the solution to the system of inequalities.
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