Solving Systems Of Inequalities: A Step-by-Step Guide
Hey guys! Today, we're diving into the world of systems of inequalities. Specifically, we're going to figure out how to determine which point, from a given set, satisfies a system of inequalities. Let's break it down step by step. We will go through each point, plugging it into the inequalities, and seeing if it satisfies both.
Understanding Systems of Inequalities
Before we jump into solving the problem, let's make sure we understand what a system of inequalities is. A system of inequalities is a set of two or more inequalities that must be satisfied simultaneously. The solution to a system of inequalities is the region in the coordinate plane that satisfies all the inequalities in the system. In simpler terms, it's the area where all the inequalities overlap. When we're given specific points, like in our problem, we need to check if those points fall within this overlapping region.
The beauty of inequalities lies in their flexibility. Unlike equations, which demand precise values, inequalities allow for a range of solutions. This makes them incredibly useful in modeling real-world scenarios where exactness is less critical than defining boundaries or limits. Think about constraints in manufacturing, budget limitations, or even the acceptable range of temperatures for a chemical reaction – inequalities are the unsung heroes behind the scenes. They enable us to define feasible regions and optimize outcomes within certain parameters.
When you graph inequalities, you're essentially creating a visual representation of these feasible regions. Each inequality carves out a specific area on the coordinate plane, and the overlapping section represents the solution set for the entire system. This graphical approach is particularly handy when dealing with more complex systems involving multiple inequalities or non-linear functions. By visualizing the solution space, you gain a deeper understanding of the relationships between the variables and the constraints that govern them. It's like having a map that guides you towards the optimal solutions within the defined boundaries.
The Problem
We're given the following system of inequalities:
- y ≥ (1/2)x² + 4
- y < √(x+2) + 6
And we need to determine which of the following points is a solution:
- A. (-2, 6)
- B. (1, 9)
- C. (2, 8)
- D. (0, 4)
Step-by-Step Solution
Let's test each point to see which one satisfies both inequalities. We will substitute the x and y coordinates of each point into both inequalities and check if the inequalities hold true.
A. (-2, 6)
First Inequality: y ≥ (1/2)x² + 4
Substitute x = -2 and y = 6:
6 ≥ (1/2)(-2)² + 4 6 ≥ (1/2)(4) + 4 6 ≥ 2 + 4 6 ≥ 6 (True)
Second Inequality: y < √(x+2) + 6
Substitute x = -2 and y = 6:
6 < √(-2+2) + 6 6 < √0 + 6 6 < 0 + 6 6 < 6 (False)
Since the point (-2, 6) satisfies the first inequality but not the second, it is not a solution to the system.
B. (1, 9)
First Inequality: y ≥ (1/2)x² + 4
Substitute x = 1 and y = 9:
9 ≥ (1/2)(1)² + 4 9 ≥ (1/2)(1) + 4 9 ≥ 0.5 + 4 9 ≥ 4.5 (True)
Second Inequality: y < √(x+2) + 6
Substitute x = 1 and y = 9:
9 < √(1+2) + 6 9 < √3 + 6 9 < 1.732 + 6 (approximately) 9 < 7.732 (False)
Since the point (1, 9) satisfies the first inequality but not the second, it is not a solution to the system.
C. (2, 8)
First Inequality: y ≥ (1/2)x² + 4
Substitute x = 2 and y = 8:
8 ≥ (1/2)(2)² + 4 8 ≥ (1/2)(4) + 4 8 ≥ 2 + 4 8 ≥ 6 (True)
Second Inequality: y < √(x+2) + 6
Substitute x = 2 and y = 8:
8 < √(2+2) + 6 8 < √4 + 6 8 < 2 + 6 8 < 8 (False)
Since the point (2, 8) satisfies the first inequality but not the second, it is not a solution to the system.
D. (0, 4)
First Inequality: y ≥ (1/2)x² + 4
Substitute x = 0 and y = 4:
4 ≥ (1/2)(0)² + 4 4 ≥ (1/2)(0) + 4 4 ≥ 0 + 4 4 ≥ 4 (True)
Second Inequality: y < √(x+2) + 6
Substitute x = 0 and y = 4:
4 < √(0+2) + 6 4 < √2 + 6 4 < 1.414 + 6 (approximately) 4 < 7.414 (True)
Since the point (0, 4) satisfies both inequalities, it is a solution to the system.
Final Answer
The point that is a solution to the system of inequalities is D. (0, 4).
Tips for Solving Systems of Inequalities
- Understand the Inequalities: Make sure you know what each inequality represents graphically. For example, y ≥ (1/2)x² + 4 represents the region above or on the parabola y = (1/2)x² + 4, while y < √(x+2) + 6 represents the region below the curve y = √(x+2) + 6.
- Test Points Methodically: When given points, substitute each point into the inequalities one at a time. This avoids confusion and ensures accuracy.
- Check Both Inequalities: A point is only a solution if it satisfies all inequalities in the system. If a point fails even one inequality, it's not a solution.
- Graphing (Optional): If you're allowed, graphing the inequalities can give you a visual representation of the solution region. This can help you confirm your algebraic solutions.
- Pay Attention to Boundary Conditions: Inequalities like y ≥ include the boundary (the line or curve itself), while inequalities like y > do not. This distinction is important when the boundary is part of the solution set.
Solving systems of inequalities might seem tricky at first, but with practice, you'll get the hang of it. The key is to take it step by step, understand what each inequality means, and carefully test each point. Keep practicing, and you'll become a pro in no time! You got this!