Determining Ordered Pairs That Satisfy Systems Of Inequalities

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In mathematics, solving systems of inequalities is a fundamental skill, particularly when dealing with linear inequalities. This article will guide you through the process of identifying an ordered pair that satisfies two given inequalities. We will delve into the mechanics of substituting values, interpreting the results, and understanding the graphical representation of these inequalities. By the end of this discussion, you'll have a solid understanding of how to tackle these types of problems efficiently and accurately.

Understanding Linear Inequalities

Linear inequalities are mathematical expressions that use inequality symbols such as '>', '>>', '<', or '<=' to compare two expressions. Unlike equations, which have definite solutions, inequalities represent a range of values. When dealing with two-variable linear inequalities, the solutions are ordered pairs (x, y) that, when substituted into the inequality, make the statement true. Graphically, the solution to a linear inequality is represented by a region in the coordinate plane, bounded by a line. This line is solid if the inequality includes '=' (i.e., '>=' or '<=') and dashed if it does not (i.e., '>' or '<').

The Significance of Ordered Pairs

An ordered pair (x, y) represents a point in the coordinate plane. In the context of inequalities, an ordered pair is a solution if it satisfies the inequality. This means that when the x and y values of the ordered pair are substituted into the inequality, the resulting statement is true. For example, consider the inequality y > x + 1. The ordered pair (1, 3) is a solution because 3 > 1 + 1 is true, while the ordered pair (0, 0) is not, because 0 > 0 + 1 is false. Understanding this basic principle is crucial when dealing with systems of inequalities, where we look for ordered pairs that satisfy all inequalities simultaneously.

Graphical Representation of Inequalities

The graphical representation of a linear inequality provides a visual understanding of its solution set. To graph a linear inequality, we first treat it as a linear equation and graph the corresponding line. For example, to graph y > -3x + 3, we first graph the line y = -3x + 3. The slope-intercept form (y = mx + b) makes this straightforward: the slope (m) is -3, and the y-intercept (b) is 3. Next, we determine whether the line should be solid or dashed. Since the inequality is '>', the line is dashed, indicating that points on the line are not solutions. Finally, we shade the region that represents the solutions. For y > -3x + 3, we shade the region above the line, as this includes all points where the y-coordinate is greater than -3x + 3.

Solving Systems of Inequalities

A system of inequalities is a set of two or more inequalities involving the same variables. The solution to a system of inequalities is the set of all ordered pairs that satisfy all inequalities in the system. Graphically, this is represented by the region where the shaded areas of all inequalities overlap. This overlapping region contains all the points that satisfy every inequality in the system, making it the solution set.

Steps to Find the Solution

To find the ordered pair that makes both inequalities true, we can follow a systematic approach. This involves understanding the individual inequalities, and then finding the common solution set. Here's a step-by-step breakdown:

  1. Understand Each Inequality: First, examine each inequality separately. Understand the slope and y-intercept of the corresponding line, and determine which region (above or below the line) represents the solution set.
  2. Graph Each Inequality: Graph each inequality on the coordinate plane. Use a dashed line for strict inequalities ('>' or '<') and a solid line for inclusive inequalities ('>=' or '<='). Shade the appropriate region for each inequality.
  3. Identify the Overlapping Region: The solution to the system of inequalities is the region where the shaded areas of all inequalities overlap. This overlapping region represents the set of all ordered pairs that satisfy all inequalities in the system.
  4. Test the Given Ordered Pairs: Substitute the x and y values of each ordered pair into both inequalities. If the ordered pair makes both inequalities true, it is a solution to the system.
  5. Verify the Solution: If you have a graph, you can also visually verify that the ordered pair lies within the overlapping region. This provides a quick check to ensure your solution is correct.

Practical Example: Solving the Given System

Let's apply these steps to the given system of inequalities:

$egin{cases} y > -3x + 3 \ y ext{≥} 2x - 2

\end{cases}$

We are given two ordered pairs to test: (1, 0) and (2, 2). We will substitute each ordered pair into both inequalities to see if they hold true.

Testing Ordered Pair (1, 0)

For the ordered pair (1, 0), we substitute x = 1 and y = 0 into both inequalities:

  • For the first inequality, y > -3x + 3: 0 > -3(1) + 3 0 > -3 + 3 0 > 0. This statement is false, so (1, 0) does not satisfy the first inequality.
  • Since (1, 0) does not satisfy the first inequality, we don't need to check the second inequality. It is already determined that (1, 0) is not a solution to the system.

Testing Ordered Pair (2, 2)

For the ordered pair (2, 2), we substitute x = 2 and y = 2 into both inequalities:

  • For the first inequality, y > -3x + 3: 2 > -3(2) + 3 2 > -6 + 3 2 > -3. This statement is true.
  • For the second inequality, y ≥ 2x - 2: 2 ≥ 2(2) - 2 2 ≥ 4 - 2 2 ≥ 2. This statement is also true.

Since (2, 2) satisfies both inequalities, it is a solution to the system. Therefore, the ordered pair (2, 2) makes both inequalities true.

Graphical Verification

To verify this graphically, we would plot both inequalities on the coordinate plane.

  • For y > -3x + 3, we draw a dashed line at y = -3x + 3 and shade the region above the line.
  • For y ≥ 2x - 2, we draw a solid line at y = 2x - 2 and shade the region above the line.
  • The overlapping region is the solution set. The point (2, 2) would fall within this overlapping region, confirming our algebraic solution.

Common Mistakes and How to Avoid Them

When solving systems of inequalities, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you approach problems more effectively and ensure accuracy. Here are some frequent errors and tips on how to avoid them:

Misinterpreting Inequality Symbols

One common mistake is misinterpreting the inequality symbols. For example, confusing '>' with '≥' or '<' with '<=' can lead to incorrect shading on the graph and incorrect solutions. It’s crucial to understand that strict inequalities ('>' and '<') do not include the points on the line, which is why we use a dashed line. Inclusive inequalities ('>=' and '<=') do include the points on the line, represented by a solid line.

  • How to Avoid It: Always double-check the inequality symbols before graphing and testing points. Use a dashed line for strict inequalities and a solid line for inclusive inequalities.

Shading the Wrong Region

Another common mistake is shading the wrong region on the graph. For inequalities like y > mx + b, the region above the line should be shaded, while for y < mx + b, the region below the line should be shaded. Similarly, for inequalities involving x, the region to the right or left of the line should be shaded accordingly.

  • How to Avoid It: Use a test point to verify your shading. Pick a point that is clearly in one region (e.g., (0, 0) if it’s not on the line) and substitute its coordinates into the inequality. If the inequality holds true, shade the region containing that point; otherwise, shade the other region.

Arithmetic Errors

Arithmetic errors when substituting values or solving equations can also lead to incorrect solutions. Simple mistakes like incorrect multiplication or addition can change the outcome significantly.

  • How to Avoid It: Double-check your calculations and substitutions. Write out each step clearly and take your time to ensure accuracy. Using a calculator can also help reduce the risk of errors.

Forgetting to Check Both Inequalities

When solving a system of inequalities, it’s essential to check that the ordered pair satisfies all inequalities in the system, not just one. An ordered pair is only a solution if it makes all inequalities true.

  • How to Avoid It: Make a checklist of all inequalities and verify that the ordered pair satisfies each one. Do not stop after checking just one inequality; ensure all conditions are met.

Not Graphing Accurately

An inaccurate graph can lead to incorrect identification of the solution region. Errors in plotting the lines or shading the regions can cause you to choose the wrong ordered pairs as solutions.

  • How to Avoid It: Use graph paper to plot points and lines accurately. Double-check the slope and y-intercept when graphing lines, and use a test point to confirm the correct shading. If possible, use graphing software or a graphing calculator to verify your hand-drawn graph.

By being mindful of these common mistakes and implementing the strategies to avoid them, you can improve your accuracy and confidence in solving systems of inequalities.

Conclusion

In conclusion, finding the ordered pair that satisfies two inequalities involves a methodical approach that includes understanding the inequalities, graphing them accurately, and testing potential solutions. By substituting the coordinates of ordered pairs into the inequalities, we can determine whether they satisfy the conditions. Graphical representation provides a visual aid in identifying the solution set, which is the region where the solutions to all inequalities overlap. Avoiding common mistakes, such as misinterpreting inequality symbols or shading the wrong region, is crucial for accurate problem-solving. With practice and a clear understanding of the steps involved, you can confidently solve systems of inequalities and identify the ordered pairs that make them true. The ordered pair (2, 2) satisfies both inequalities, making it the correct solution, while (1, 0) does not satisfy the first inequality and is therefore not a solution to the system.