Verified Answer :
- Analyze the boundary lines: \(y = 0.8x - 0.2\) and \(y = 2x + 6\).
- Calculate the intersection point: \((-\frac{31}{6}, -\frac{13}{3}) \approx (-5.17, -4.33)\).
- Identify the solution region: The area above the line \(y = 0.8x - 0.2\) and below the line \(y = 2x + 6\).
- Final interpretation: The intersection creates a wedge-shaped region that narrows to the left at the point \((-\frac{31}{6}, -\frac{13}{3})\) and extends infinitely to the right.
Explanation
- Analyzing the Boundary Lines
To analyze the system of inequalities \(y \geq \frac{4}{5}x - \frac{1}{5}\) and \(y \leq 2x + 6\), we identify the characteristics of their boundary lines.
The first inequality, \(y \geq \frac{4}{5}x - \frac{1}{5}\), corresponds to a line with a slope of \(\frac{4}{5} = 0.8\) and a y-intercept of \(-\frac{1}{5} = -0.2\). Because of the \(\geq\) sign, the region includes the line itself and the area above it.
The second inequality, \(y \leq 2x + 6\), corresponds to a line with a slope of \(2\) and a y-intercept of \(6\). Because of the \(\leq\) sign, the region includes the line itself and the area below it.
- Finding the Intersection Point
To find where these lines cross, we solve the system of equations by setting the two lines equal to each other:
\[\frac{4}{5}x - \frac{1}{5} = 2x + 6\]
To solve for \(x\), we multiply the entire equation by 5 to clear the fractions:
\[4x - 1 = 10x + 30\]
Subtracting \(4x\) from both sides gives \(-1 = 6x + 30\). Subtracting 30 from both sides, we get \(-31 = 6x\), so \(x = -\frac{31}{6} \approx -5.17\).
Now, substitute \(x\) back into the second equation to find \(y\):
\[y = 2\left(-\frac{31}{6}\right) + 6 = -\frac{31}{3} + \frac{18}{3} = -\frac{13}{3} \approx -4.33\]
The lines intersect at the point \((-\frac{31}{6}, -\frac{13}{3})\).
- Characterizing the Shaded Region
The solution to the system is the intersection of two half-planes. The first line (\(y = 0.8x - 0.2\)) has a shallower positive slope, and the second line (\(y = 2x + 6\)) has a steeper positive slope. The region of interest is the wedge-shaped area that is simultaneously above the first line and below the second line. This region extends infinitely to the right, narrowing toward the intersection point \((-\frac{31}{6}, -\frac{13}{3})\). When looking for the correct graph, identify the region bounded by these two lines where the area above the less-steep line and below the more-steep line is shaded.
Examples
Systems of inequalities are used in linear programming to model constraints on resources. For example, if a small business owner has two products, A and B, they might have a budget constraint represented by one inequality and a production capacity constraint represented by another. The region of valid production combinations is the intersection of these constraints, helping the owner find the optimal mix that maximizes profit within those bounds. Visualizing these constraints as intersecting half-planes allows for a clear overview of all feasible production strategies.