- Manipulate the first inequality \(4x - 5y \leq 1\) to isolate \(y\), resulting in \(y \geq \frac{4}{5}x - \frac{1}{5}\).
- Manipulate the second inequality \(\frac{1}{2}y - x \leq 3\) to isolate \(y\), resulting in \(y \leq 2x + 6\).
- The simplified system is \(y \geq \frac{4}{5}x - \frac{1}{5}\) and \(y \leq 2x + 6\).
- Final system in \(y\) terms: \(\boxed{y \geq \frac{4}{5}x - \frac{1}{5}, y \leq 2x + 6}\).
Explanation
- Analysis and Manipulation of the System of Inequalities
The given system of linear inequalities is:
1) \(4x - 5y \leq 1\)
2) \(\frac{1}{2}y - x \leq 3\)
We aim to express these inequalities in terms of \(y\). We will perform algebraic manipulations to isolate \(y\) for each inequality.
First, consider the inequality \(4x - 5y \leq 1\). To solve for \(y\), we subtract \(4x\) from both sides:
\(-5y \leq 1 - 4x\)
Next, divide both sides by \(-5\). Because we are dividing by a negative number, we must reverse the inequality symbol:
\(y \geq \frac{1 - 4x}{-5}\)
\(y \geq -\frac{1}{5} + \frac{4}{5}x\)
\(y \geq \frac{4}{5}x - \frac{1}{5}\)
Now, consider the inequality \(\frac{1}{2}y - x \leq 3\). To solve for \(y\), we add \(x\) to both sides:
\(\frac{1}{2}y \leq x + 3\)
Next, multiply both sides by \(2\):
\(y \leq 2x + 6\)
Thus, the system is equivalent to:
\(y \geq \frac{4}{5}x - \frac{1}{5}\)
\(y \leq 2x + 6\)
Comparing this to the provided options, we note that none of the options A, B, or C perfectly match the derived inequality symbols (\(\geq\) and \(\leq\)). Assuming standard convention where the boundary conditions are equivalent, we look for the option that represents the closest match. However, strictly speaking, the derived system is \(y \geq \frac{4}{5}x - \frac{1}{5}\) and \(y \leq 2x + 6\). Given the options provided, there is no exact equivalent option among A, B, or C using standard mathematical notation for inequalities. If forced to choose the most likely intended answer in a scenario where strict inequality signs are used for non-strict systems, the logic derived remains the correct analytical approach.
- Final Conclusion
Given the steps above, we have derived the system:
\[y \geq \frac{4}{5}x - \frac{1}{5}\]
\[y \leq 2x + 6\]
This result defines the region bounded by two lines in the coordinate plane. Each line represents the boundary of an half-plane. The final solution is the intersection of these two half-planes. Since the options provided in the prompt contain strict inequalities, it is important to recognize that the derivation relies on the properties of linear inequalities and the rules for rearranging them. The conclusion is that the system can be written as \(y \geq \frac{4}{5}x - \frac{1}{5}\) and \(y \leq 2x + 6\).
Examples
Linear inequalities are used extensively in linear programming to determine feasible regions for optimization. For example, a factory might have constraints on labor hours (inequality 1) and raw materials (inequality 2) required to produce two types of products. The system of inequalities helps the factory manager identify the set of all possible production plans (the feasible region) that satisfy all constraints simultaneously, ensuring efficient resource allocation. Graphing these boundaries on a coordinate plane allows for a clear visual representation of where these conditions are met.