Ordered Pair Solution For Inequalities A Step-by-Step Guide

When dealing with systems of inequalities, finding the ordered pair that satisfies all inequalities can be a crucial task. This article will delve into a step-by-step method to identify such ordered pairs, using the example inequalities:

$y < 3x - 1$ $y \geq -x + 4$

We will analyze each provided ordered pair to determine if it satisfies both inequalities. This approach ensures a clear understanding of how to solve similar problems efficiently.

Understanding Linear Inequalities

Before diving into the solution, it's important to grasp the concept of linear inequalities. Linear inequalities are mathematical expressions that compare two quantities using inequality symbols such as < (less than), > (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to). Unlike linear equations, which have a single solution, linear inequalities have a range of solutions. When dealing with two-variable inequalities, the solutions are often represented as a region on the coordinate plane.

To determine if an ordered pair (x, y) is a solution to a system of inequalities, we substitute the x and y values into each inequality. If the ordered pair makes all the inequalities true, then it is a solution to the system. This process involves careful arithmetic and a clear understanding of inequality symbols.

For instance, the inequality $y < 3x - 1$ represents all the points (x, y) that lie below the line $y = 3x - 1$. Similarly, the inequality $y \geq -x + 4$ represents all the points (x, y) that lie on or above the line $y = -x + 4$. The solution to the system of these two inequalities is the region where these two individual solution sets overlap.

Understanding these basics is crucial for tackling the problem at hand, where we need to identify which ordered pair, among the given options, satisfies both inequalities simultaneously. This foundational knowledge sets the stage for a methodical approach to solving the problem.

Step-by-Step Solution

To determine which ordered pair satisfies both inequalities, we will evaluate each option individually. This methodical approach ensures accuracy and clarity in our solution.

Option A: (4, 0)

Let's substitute x = 4 and y = 0 into the inequalities:

1. $y < 3x - 1$ $0 < 3(4) - 1$ $0 < 12 - 1$ $0 < 11$ (True)
2. $y \geq -x + 4$ $0 \geq -(4) + 4$ $0 \geq -4 + 4$ $0 \geq 0$ (True)

Since (4, 0) satisfies both inequalities, it is a potential solution. However, we must check the other options to ensure we find the correct answer.

Option B: (1, 2)

Substitute x = 1 and y = 2 into the inequalities:

1. $y < 3x - 1$ $2 < 3(1) - 1$ $2 < 3 - 1$ $2 < 2$ (False)

Since 2 is not less than 2, this inequality is not satisfied. Therefore, (1, 2) is not a solution to the system of inequalities. We can stop evaluating the second inequality for this ordered pair, as both must be true for the ordered pair to be a solution.

Option C: (0, 4)

Substitute x = 0 and y = 4 into the inequalities:

1. $y < 3x - 1$ $4 < 3(0) - 1$ $4 < 0 - 1$ $4 < -1$ (False)

Again, since 4 is not less than -1, this inequality is not satisfied. Thus, (0, 4) is not a solution to the system of inequalities. We can disregard this option as well.

Option D: (2, 1)

Substitute x = 2 and y = 1 into the inequalities:

1. $y < 3x - 1$ $1 < 3(2) - 1$ $1 < 6 - 1$ $1 < 5$ (True)
2. $y \geq -x + 4$ $1 \geq -(2) + 4$ $1 \geq -2 + 4$ $1 \geq 2$ (False)

In this case, the first inequality is satisfied, but the second inequality is not, as 1 is not greater than or equal to 2. Therefore, (2, 1) is not a solution to the system of inequalities.

Final Answer

After evaluating each option, we found that only option A, (4, 0), satisfies both inequalities. This methodical approach ensures that we have thoroughly checked each possibility, leading to the correct solution.

Graphical Interpretation

Visualizing inequalities on a graph provides an intuitive understanding of the solution set. Each inequality represents a region on the coordinate plane, and the solution to the system of inequalities is the region where these individual regions overlap.

To graph the inequality $y < 3x - 1$, we first graph the line $y = 3x - 1$. This is a straight line with a slope of 3 and a y-intercept of -1. Since the inequality is strictly less than, we draw the line as a dashed line to indicate that the points on the line are not included in the solution. The solution to the inequality is the region below this dashed line. This is because for any x-value, the y-values that satisfy the inequality are those that are less than the corresponding y-value on the line.

Similarly, for the inequality $y \geq -x + 4$, we graph the line $y = -x + 4$. This is a straight line with a slope of -1 and a y-intercept of 4. Since the inequality includes "equal to", we draw the line as a solid line to indicate that the points on the line are included in the solution. The solution to the inequality is the region above this solid line. This is because for any x-value, the y-values that satisfy the inequality are those that are greater than or equal to the corresponding y-value on the line.

The solution to the system of inequalities is the intersection of these two regions. This is the area on the coordinate plane where the shaded regions of both inequalities overlap. The ordered pair (4, 0) lies within this overlapping region, confirming that it is a solution to the system. The other ordered pairs, (1, 2), (0, 4), and (2, 1), do not lie within this region, which aligns with our algebraic findings.

The graphical approach not only confirms the algebraic solution but also provides a visual representation of the solution set. This is particularly useful for understanding systems of inequalities and their solutions in a more comprehensive manner. By plotting the inequalities on a graph, one can quickly identify the region that satisfies all the inequalities, making it a valuable tool in problem-solving.

Common Mistakes to Avoid

When solving systems of inequalities, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help ensure accuracy and efficiency in problem-solving.

Incorrect Substitution

One frequent error is incorrect substitution of x and y values into the inequalities. This can happen due to simple arithmetic mistakes or misreading the values. To avoid this, always double-check the values before substituting them and perform the calculations carefully. It is a good practice to write down each step clearly to minimize errors.

Misinterpreting Inequality Symbols

Misinterpreting inequality symbols is another common mistake. For example, confusing < (less than) with $\leq$ (less than or equal to) can lead to including or excluding boundary points incorrectly. Remember that < and > do not include the boundary line, while $\leq$ and $\geq$ do. When graphing, use a dashed line for strict inequalities (< and >) and a solid line for inequalities that include equality ($\leq$ and $\geq$).

Only Checking One Inequality

Another mistake is only checking one inequality in the system. An ordered pair must satisfy all inequalities in the system to be a solution. If an ordered pair fails to satisfy even one inequality, it is not a solution to the system. Ensure you substitute the values into each inequality and verify that they all hold true.

Algebraic Errors

Algebraic errors during simplification can also lead to incorrect conclusions. These errors might include mistakes in multiplying, dividing, adding, or subtracting terms. To prevent these errors, it is helpful to simplify each inequality step by step, showing all your work. This makes it easier to spot and correct any mistakes.

Conclusion

In summary, solving systems of inequalities requires careful attention to detail and a methodical approach. By avoiding these common mistakes and following the steps outlined in this article, you can confidently determine which ordered pairs satisfy a given set of inequalities. Remember to double-check your work, especially when substituting values and simplifying expressions. A clear understanding of the graphical representation of inequalities can also provide a visual check for your solutions, ensuring accuracy and a deeper understanding of the concepts involved.

Practice Problems

To solidify your understanding of solving systems of inequalities, working through practice problems is essential. Here are a few problems that will allow you to apply the methods discussed in this article. Try solving them on your own, and then check your answers against the solutions provided.

Problem 1

Which ordered pair makes both inequalities true?

$y > 2x + 1$ $y \leq -x + 5$

A. (0, 0) B. (1, 4) C. (2, -1) D. (-1, 2)

Problem 2

Identify the ordered pair that satisfies the following system of inequalities:

$y \geq 3x - 2$ $y < -2x + 4$

A. (0, -3) B. (1, 1) C. (2, 2) D. (-1, 5)

Problem 3

Which of the following ordered pairs is a solution to the system of inequalities?

$y < -x + 3$ $y > x - 1$

A. (0, 4) B. (2, 1) C. (3, 0) D. (-1, -2)

Solutions

Solution to Problem 1

• A. (0, 0)

  • $0 > 2(0) + 1$ → $0 > 1$ (False) Since the first inequality is false, (0, 0) is not a solution.

• B. (1, 4)

  • $4 > 2(1) + 1$ → $4 > 3$ (True)
  • $4 \leq -(1) + 5$ → $4 \leq 4$ (True) (1, 4) satisfies both inequalities.

• C. (2, -1)

  • $-1 > 2(2) + 1$ → $-1 > 5$ (False) Since the first inequality is false, (2, -1) is not a solution.

• D. (-1, 2)

  • $2 > 2(-1) + 1$ → $2 > -1$ (True)
  • $2 \leq -(-1) + 5$ → $2 \leq 6$ (True) (-1, 2) satisfies both inequalities.

  Correct Answer: B. (1, 4)

Solution to Problem 2

• A. (0, -3)

  • $-3 \geq 3(0) - 2$ → $-3 \geq -2$ (False) Since the first inequality is false, (0, -3) is not a solution.

• B. (1, 1)

  • $1 \geq 3(1) - 2$ → $1 \geq 1$ (True)
  • $1 < -2(1) + 4$ → $1 < 2$ (True) (1, 1) satisfies both inequalities.

• C. (2, 2)

  • $2 \geq 3(2) - 2$ → $2 \geq 4$ (False) Since the first inequality is false, (2, 2) is not a solution.

• D. (-1, 5)

  • $5 \geq 3(-1) - 2$ → $5 \geq -5$ (True)
  • $5 < -2(-1) + 4$ → $5 < 6$ (True) (-1, 5) satisfies both inequalities.

  Correct Answer: B. (1, 1)

Solution to Problem 3

• A. (0, 4)

  • $4 < -(0) + 3$ → $4 < 3$ (False) Since the first inequality is false, (0, 4) is not a solution.

• B. (2, 1)

  • $1 < -(2) + 3$ → $1 < 1$ (False) Since the first inequality is false, (2, 1) is not a solution.

• C. (3, 0)

  • $0 < -(3) + 3$ → $0 < 0$ (False) Since the first inequality is false, (3, 0) is not a solution.

• D. (-1, -2)

  • $-2 < -(-1) + 3$ → $-2 < 4$ (True)
  • $-2 > (-1) - 1$ → $-2 > -2$ (False) Since the second inequality is false, (-1, -2) is not a solution.

  Correct Answer: None of the above

These practice problems provide a range of scenarios for you to apply the techniques discussed in this article. By working through these examples and checking your answers, you can strengthen your understanding of how to solve systems of inequalities and avoid common mistakes. Remember to approach each problem methodically, substitute values carefully, and double-check your work to ensure accuracy.