Solve the system of inequalities:
[tex]
\begin{array}{l}
y \leq 2 x \\
y \geq-2
\end{array}
[/tex]
Answer (1)
Graph the inequality y ≤ 2 x , which is the region below the line y = 2 x .
Graph the inequality y g e q − 2 , which is the region above the line y = − 2 .
Find the intersection of the two regions, which is the solution to the system of inequalities.
The solution is the region bounded by the lines y = 2 x and y = − 2 .
Explanation
Understanding the Problem We are given a system of two inequalities:
y ≤ 2 x
y g e q − 2
We need to find the region in the xy-plane that satisfies both inequalities simultaneously.
Solving the System of Inequalities To solve this system of inequalities, we need to find the region in the xy-plane where both inequalities are true. We will graph each inequality separately and then find the intersection of the two regions.
Graphing the First Inequality First, let's consider the inequality y ≤ 2 x . This inequality represents the region below the line y = 2 x . To graph this line, we can find two points on the line. For example, when x = 0 , y = 2 ( 0 ) = 0 , so ( 0 , 0 ) is a point on the line. When x = 1 , y = 2 ( 1 ) = 2 , so ( 1 , 2 ) is another point on the line. We can draw a line through these two points. Since the inequality is y ≤ 2 x , we shade the region below the line.
Graphing the Second Inequality Next, let's consider the inequality y g e q − 2 . This inequality represents the region above the horizontal line y = − 2 . To graph this line, we draw a horizontal line at y = − 2 . Since the inequality is y g e q − 2 , we shade the region above the line.
Finding the Solution Region The solution to the system of inequalities is the intersection of the two shaded regions. This is the region bounded below by the line y = − 2 and above by the line y = 2 x .
Finding the Intersection Point To find the intersection point of the two lines y = 2 x and y = − 2 , we can substitute y = − 2 into y = 2 x to get − 2 = 2 x , so x = − 1 . The intersection point is ( − 1 , − 2 ) .
Describing the Solution Region The solution is the set of all points ( x , y ) such that y g e q − 2 and y ≤ 2 x . This region extends infinitely to the right.
Final Answer The solution to the system of inequalities is the region bounded by the lines y = 2 x and y = − 2 .
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, where you want to optimize a certain objective function subject to constraints. For example, a company might want to maximize its profit given constraints on resources like labor and materials. The feasible region, representing all possible solutions that satisfy the constraints, is defined by a system of inequalities. By finding the vertices of this region, the company can determine the optimal production levels to maximize profit. Another application is in diet planning, where you want to meet certain nutritional requirements (e.g., minimum intake of vitamins and minerals) while staying within a budget. The constraints on nutrient intake and budget can be expressed as a system of inequalities, and the feasible region represents all possible diets that meet the requirements.
