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, representing 2x + 4"> y > 2 x + 4 .
Graph the line x + y = 6 as a solid line and shade the region below it, representing x + y ≤ 6 .
Find the intersection point of the two lines by solving the system of equations, resulting in the point ( 3 2 , 3 16 ) .
The solution set is the intersection of the two shaded regions, bounded by the two lines. See graph for solution
Explanation
Understanding the Problem We are given a system of two inequalities:
2x + 4"> y > 2 x + 4
x + y ≤ 6
Our goal is to graph the solution set of this system on the coordinate plane. This means we need to find all the points ( x , y ) that satisfy both inequalities simultaneously.
Analyzing the First Inequality First, let's analyze the first inequality, 2x + 4"> y > 2 x + 4 . This inequality represents the region above the line y = 2 x + 4 . To graph this, we first graph the line y = 2 x + 4 as a dashed line because the inequality is strict (i.e., it does not include the equals sign). Then, we shade the region above the line.
Analyzing the Second Inequality Next, let's analyze the second inequality, x + y ≤ 6 . This inequality represents the region on and below the line x + y = 6 . We can rewrite this as y ≤ − x + 6 . To graph this, we graph the line y = − x + 6 as a solid line because the inequality includes equality. Then, we shade the region below the line.
Finding the Intersection Point The solution set of the system of inequalities is the intersection of the two shaded regions. To find the intersection point of the two lines, we set the expressions for y equal to each other:
2 x + 4 = − x + 6
Adding x to both sides gives:
3 x + 4 = 6
Subtracting 4 from both sides gives:
3 x = 2
Dividing by 3 gives:
x = 3 2
Now, we substitute this value of x back into either equation to find the corresponding value of y . Let's use the second equation:
y = − 3 2 + 6 = − 3 2 + 3 18 = 3 16
So, the intersection point is ( 3 2 , 3 16 ) .
Determining the Solution Set The solution to the system of inequalities is the region where the shading from both inequalities overlaps. The first inequality 2x + 4"> y > 2 x + 4 is satisfied above the dashed line y = 2 x + 4 , and the second inequality x + y ≤ 6 is satisfied below the solid line x + y = 6 . The intersection point of the two lines is ( 3 2 , 3 16 ) .
Final Answer Therefore, the solution is the area bounded by the lines y = 2 x + 4 and x + y = 6 , where 2x + 4"> y > 2 x + 4 and x + y ≤ 6 .
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 nutrition to plan diets that meet certain nutritional requirements within budget limits. For instance, a system of inequalities could represent the constraints on the amount of carbohydrates and fats a person can consume daily while also staying within a certain calorie range. Graphing the solution set helps visualize all possible combinations of food intake that satisfy these constraints, aiding in making informed dietary choices.
To graph the solution of the system of inequalities 2x + 4"> y > 2 x + 4 and x + y ≤ 6 , plot the line y = 2 x + 4 as a dashed line and shade above it, and plot x + y = 6 as a solid line and shade below it. The solution set is where these shaded regions overlap, including the intersection point ( 3 2 , 3 16 ) .
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