Use the drawing tools to graph the solution to this system of inequalities on the coordinate plane.
[tex]
\begin{array}{l}
y\ \textgreater \ 2 x+4 \
x+y \leq 6
\end{array}
[/tex]
Answer (2)
Graph the line y = 2 x + 4 as a dashed line and shade the region above it since 2x + 4"> y > 2 x + 4 .
Graph the line x + y = 6 as a solid line and shade the region below it since x + y ≤ 6 .
The solution to the system of inequalities is the intersection of the two shaded regions.
The graph represents the solution set to the system of inequalities.
Explanation
Analyze the problem We are given a system of two inequalities:
2x + 4"> y > 2 x + 4
$x + y
We need to graph the solution to this system of inequalities on the coordinate plane.
Analyze the first inequality First, let's analyze the first inequality: 2x + 4"> y > 2 x + 4 . This is a linear inequality. To graph it, we first graph the corresponding line y = 2 x + 4 . The slope of this line is 2 and the y-intercept is 4. Since the inequality is strict ( "> > ), we draw a dashed line. To determine which side of the line to shade, we can test a point, say (0, 0). Plugging this into the inequality, we get 2(0) + 4"> 0 > 2 ( 0 ) + 4 , which simplifies to 4"> 0 > 4 . This is false, so we shade the region above the dashed line.
Analyze the second inequality Next, let's analyze the second inequality: $x + y
To determine which side of the line to shade, we can test a point, say (0, 0). Plugging this into the inequality, we get $0 + 0
So, we shade the region below the solid line.
Find the solution The solution to the system of inequalities is the region where the two shaded regions overlap.
Examples
Systems of inequalities are used in various real-world applications, such as in economics to determine feasible production regions given resource constraints, or in optimization problems to find the best solution within a set of limitations. For example, a company might use a system of inequalities to determine the optimal number of products to produce given constraints on labor, materials, and budget. Graphing these inequalities helps visualize the feasible region and identify potential solutions.
To graph the system of inequalities, first graph the line for 2x + 4"> y > 2 x + 4 as a dashed line and shade above it. Then, graph the line for x + y ≤ 6 as a solid line and shade below it. The solution is the overlap of the shaded regions from both inequalities.
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