When solving for ordered pairs that satisfy multiple inequalities, it’s important to understand the relationship between the inequalities and how to find the common solution. In this article, we will explore the process of finding the ordered pair that makes both inequalities true, and discuss various scenarios to provide a comprehensive understanding of the topic.
Understanding Inequalities
Inequalities are mathematical expressions that compare two quantities and indicate their relationship using symbols such as “<", "<=", ">“, “>=”, or “<>“. When solving for ordered pairs that satisfy multiple inequalities, we need to consider the common solution that satisfies all the given conditions simultaneously.
Finding the Common Solution
When dealing with multiple inequalities, finding the common solution involves identifying the region of the coordinate plane that satisfies all the given conditions. This can be done by graphing the inequalities and determining the overlapping region, or by solving the inequalities algebraically to find the common solution.
Graphing the Inequalities: To graph the inequalities, plot the boundary lines and shade the regions that satisfy each individual inequality. The overlapping shaded region represents the common solution that satisfies all the given conditions.
Solving Algebraically: To solve the inequalities algebraically, start by solving each inequality separately to find the range of values that satisfy it. Then, identify the overlapping range of values that satisfies all the given conditions, which represents the common solution.
Scenario Analysis
Let’s consider a scenario where we have the following two inequalities:
- 2x + 3y <= 12
- x – y > 1
We will determine the ordered pair that makes both inequalities true using both graphical and algebraic methods to illustrate the process.
Graphical Method
To find the common solution graphically, we can plot the boundary lines of the inequalities and shade the regions that satisfy each individual inequality. The overlapping shaded region represents the common solution.
Step 1: Graph the inequality 2x + 3y <= 12
The boundary line for this inequality is 2x + 3y = 12. Plot the line and shade the region below it, including the line since the inequality includes the equal sign.
Step 2: Graph the inequality x – y > 1
The boundary line for this inequality is x – y = 1. Plot the line and shade the region above it, excluding the line since the inequality does not include the equal sign.
Step 3: Determine the overlapping region
The overlapping region between the two inequalities represents the common solution that satisfies both conditions simultaneously. The ordered pair within this region will make both inequalities true.
Algebraic Method
To find the common solution algebraically, we can solve each inequality separately to find the range of values that satisfy it, and then identify the overlapping range of values that satisfies both inequalities.
Step 1: Solve the inequality 2x + 3y <= 12 for y
Subtract 2x from both sides to get 3y <= -2x + 12. Then, divide by 3 to get y <= (-2/3)x + 4.
Step 2: Solve the inequality x – y > 1 for y
Subtract x from both sides to get -y > 1 – x. Then, multiply by -1 and reverse the inequality sign to get y < x - 1.
Step 3: Determine the overlapping range of values for y
By comparing the two inequalities, we can identify the overlapping range of values for y that satisfies both conditions. This overlapping range represents the common solution.
Conclusion
When solving for the ordered pair that makes both inequalities true, it’s essential to consider the relationship between the inequalities and find the common solution that satisfies all the given conditions simultaneously. Whether using the graphical method or the algebraic method, understanding the concepts of inequalities and their graphical representations is key to finding the common solution accurately.
