Verified Answer :

  • Simplify the first inequality to the slope-intercept form \(y \geq \frac{1}{4}x - 1\), which defines a solid boundary line with shading above it.
  • Simplify the second inequality to \(y < x + 1\), which defines a dashed boundary line with shading below it.
  • Determine that the solution region is the intersection of these two areas, formed between the solid and dashed lines.
  • The final solution region is the intersection: \(\{(x, y) \mid y \geq \frac{1}{4}x - 1 \text{ and } y < x + 1\}\).

Explanation

  1. Problem Analysis

The problem requires identifying the region defined by the system of two linear inequalities:
1) \(x - 4y \leq 4\)
2) \(y < x + 1\)
To visualize the solution, we first determine the equations of the boundary lines and then identify the shading for each inequality. The solution is the intersection of these two regions.

  1. Analyzing the First Inequality

The first inequality is \(x - 4y \leq 4\). To find the boundary line, we treat it as an equation: \(x - 4y = 4\). Rearranging for \(y\) gives:
\[-4y = -x + 4 \implies y = \frac{1}{4}x - 1.\]
Since the inequality is inclusive (\(\leq\)), the boundary line is solid. To determine the shading, we can test the origin \((0,0)\):
\(0 - 4(0) \leq 4 \implies 0 \leq 4\), which is true. Therefore, the region includes the origin, meaning the shading is above the solid line \(y = \frac{1}{4}x - 1\).

  1. Analyzing the Second Inequality

The second inequality is \(y < x + 1\). The boundary line is \(y = x + 1\). Since the inequality is strict (\(<\)), the boundary line is dashed. Again, testing the origin \((0,0)\):
\(0 < 0 + 1 \implies 0 < 1\), which is true. The region includes the origin, meaning the shading is below the dashed line \(y = x + 1\).

  1. Conclusion on the Solution Region

Combining the two results:
1) The boundary lines are \(y = \frac{1}{4}x - 1\) (solid) and \(y = x + 1\) (dashed).
2) The solution region is the area that lies above the solid line \(y = \frac{1}{4}x - 1\) and simultaneously below the dashed line \(y = x + 1\).
These two lines intersect at a point found by setting \(\frac{1}{4}x - 1 = x + 1\):
\[-1 - 1 = x - \frac{1}{4}x \implies -2 = \frac{3}{4}x \implies x = -\frac{8}{3}.\]
Substituting \(x = -\frac{8}{3}\) into \(y = x + 1\):
\[y = -\frac{8}{3} + 1 = -\frac{5}{3}.\]
The intersection point is \((-\frac{8}{3}, -\frac{5}{3})\). The solution region is an infinite, wedge-shaped area to the right of this intersection point.

Examples

Understanding systems of inequalities is essential in optimization, such as supply chain management or production planning. For example, a company producing two products must keep production within capacity constraints (e.g., raw materials, machine hours) defined by linear inequalities. Graphing these constraints allows managers to identify the "feasible region"—the range of production levels where operations remain profitable and within resource limits. By visualizing this, they can make informed decisions to maximize efficiency.

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