Solving $2x + 3 > 7$ And $x + 9 < 16$ A Step-by-Step Guide

Understanding and solving inequalities is a fundamental skill in mathematics, with applications ranging from basic algebra to advanced calculus. Inequalities help us define ranges and boundaries, rather than specific values, and are crucial in various fields like economics, engineering, and computer science. In this article, we will delve into the process of solving a compound inequality, specifically focusing on the example $2x + 3 > 7$ and $x + 9 < 16$. We will break down each step, explain the underlying principles, and provide a clear, comprehensive guide to help you master this essential mathematical concept.

Introduction to Inequalities

Before we tackle the specific problem, let's establish a solid understanding of what inequalities are and how they differ from equations. An inequality is a mathematical statement that compares two expressions using inequality symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Unlike equations, which have a single solution or a finite set of solutions, inequalities often have a range of solutions. This range represents all the values that satisfy the inequality condition.

For instance, the inequality $x > 3$ means that $x$ can be any number greater than 3, but not 3 itself. Similarly, $x ≤ 5$ indicates that $x$ can be any number less than or equal to 5. Understanding these symbols and the concepts they represent is the first step in solving inequalities. Inequalities play a pivotal role in various mathematical and real-world scenarios. They are used to define constraints in optimization problems, determine feasible regions in linear programming, and model real-life situations where exact values are not necessary or possible. For example, in economics, inequalities can be used to represent budget constraints or supply and demand relationships. In engineering, they might describe tolerance levels for measurements or the range of operating conditions for a system.

Understanding Compound Inequalities

A compound inequality is essentially two or more inequalities joined together by the words “and” or “or.” These connecting words are crucial because they determine how we interpret and solve the inequalities. When inequalities are joined by “and,” it means that both inequalities must be true simultaneously. The solution set is the intersection of the individual solution sets. On the other hand, when inequalities are joined by “or,” it means that at least one of the inequalities must be true. The solution set is the union of the individual solution sets. In our case, we are dealing with a compound inequality connected by “and”: $2x + 3 > 7$ and $x + 9 < 16$. This means we need to find the values of $x$ that satisfy both inequalities at the same time. The word “and” imposes a stricter condition on the solution set, as it requires a value to meet two different criteria. This is different from an “or” scenario, where a value only needs to satisfy one of the conditions.

Solving a compound inequality involves solving each inequality separately and then combining the solutions based on the connecting word. For “and” inequalities, we look for the overlap in the solutions; for “or” inequalities, we combine all solutions. This process requires careful attention to detail and a solid understanding of basic algebraic principles. The significance of compound inequalities extends beyond the classroom. In practical applications, they are used to model scenarios where multiple conditions must be met. For example, a manufacturing process might require a temperature to be within a certain range (an “and” condition), or a product might meet quality standards if it passes either of two different tests (an “or” condition).

Solving the First Inequality: $2x + 3 > 7$

Let’s begin by solving the first inequality, $2x + 3 > 7$. Our goal is to isolate $x$ on one side of the inequality. To do this, we follow a similar process to solving equations, but with a crucial difference: when we multiply or divide both sides by a negative number, we must reverse the inequality sign. The first step in isolating $x$ is to subtract 3 from both sides of the inequality. This operation maintains the balance of the inequality and helps us move closer to isolating the variable. Subtracting 3 from both sides, we get:

$2x + 3 - 3 > 7 - 3$

This simplifies to:

$2x > 4$

Now, we need to get $x$ by itself. To do this, we divide both sides of the inequality by 2. Since 2 is a positive number, we do not need to reverse the inequality sign. Dividing both sides by 2, we have:

rac{2x}{2} > rac{4}{2}

This simplifies to:

$x > 2$

So, the solution to the first inequality is $x > 2$. This means that any value of $x$ greater than 2 will satisfy the condition $2x + 3 > 7$. It's essential to understand that this is not a single solution but a range of solutions. This range can be visualized on a number line as all points to the right of 2, excluding 2 itself (since the inequality is strictly greater than).

Solving the Second Inequality: $x + 9 < 16$

Next, we will solve the second inequality, $x + 9 < 16$. Again, our objective is to isolate $x$ on one side of the inequality. In this case, we need to eliminate the +9 term on the left side. We can accomplish this by subtracting 9 from both sides of the inequality. This operation preserves the inequality and helps us isolate the variable. Subtracting 9 from both sides, we get:

$x + 9 - 9 < 16 - 9$

This simplifies to:

$x < 7$

Therefore, the solution to the second inequality is $x < 7$. This means that any value of $x$ less than 7 will satisfy the condition $x + 9 < 16$. Similar to the first inequality, this is a range of solutions. On a number line, this solution set is represented by all points to the left of 7, not including 7 itself (as the inequality is strictly less than). Understanding the solutions to each individual inequality is a crucial step in solving the compound inequality. It allows us to identify the ranges of values that satisfy each condition separately, which we then need to combine based on the connecting word (“and” in this case).

Combining the Solutions: “And” Condition

Now that we have solved both inequalities individually, we need to combine their solutions based on the “and” condition. Recall that “and” means that both inequalities must be true simultaneously. We found that the solution to the first inequality, $2x + 3 > 7$, is $x > 2$, and the solution to the second inequality, $x + 9 < 16$, is $x < 7$. To find the solution to the compound inequality, we need to determine the values of $x$ that satisfy both $x > 2$ and $x < 7$. This is where the concept of intersection comes into play. We are looking for the overlap between the two solution sets. To visualize this, we can imagine two number lines. On the first number line, we shade all values greater than 2. On the second number line, we shade all values less than 7. The solution to the compound inequality is the region where the shaded areas overlap. This overlap represents the values that are both greater than 2 and less than 7.

Mathematically, we can express this combined solution as the interval $2 < x < 7$. This notation means that $x$ is greater than 2 and less than 7. It is a concise way to represent the range of values that satisfy the compound inequality. This range does not include 2 or 7 themselves, as the original inequalities were strict (greater than and less than, not greater than or equal to and less than or equal to). Understanding how to combine solutions based on the “and” condition is essential for solving compound inequalities. It requires careful consideration of the individual solution sets and how they intersect. This concept is also fundamental in various mathematical applications, such as finding the domain of functions or solving systems of inequalities.

Representing the Solution on a Number Line

A number line is a powerful tool for visualizing the solutions to inequalities, especially compound inequalities. It provides a clear and intuitive way to understand the range of values that satisfy the given conditions. To represent the solution $2 < x < 7$ on a number line, we first draw a horizontal line and mark the key points, which in this case are 2 and 7. Since the inequality is strict (i.e., $x$ is strictly greater than 2 and strictly less than 7), we use open circles at 2 and 7. An open circle indicates that the endpoint is not included in the solution set. If the inequalities were non-strict (i.e., $x$ is greater than or equal to 2 or less than or equal to 7), we would use closed circles to indicate that the endpoints are included. Next, we shade the region between 2 and 7. This shaded region represents all the values of $x$ that satisfy the compound inequality. Any point within this shaded region, excluding 2 and 7, is a solution to the problem. The number line representation provides a visual confirmation of the algebraic solution. It clearly shows the range of values that satisfy both inequalities simultaneously. This visual aid is particularly helpful for students learning about inequalities, as it bridges the gap between abstract algebraic concepts and concrete graphical representations. Moreover, the number line representation is a valuable tool in more advanced mathematical contexts, such as calculus and real analysis, where understanding intervals and ranges is crucial.

Importance of Checking the Solution

After solving any mathematical problem, it's crucial to check the solution to ensure accuracy. This is especially important with inequalities, where a small error can significantly alter the solution set. To check our solution $2 < x < 7$, we can pick a value within this range and substitute it into the original inequalities. For example, let's choose $x = 4$, which falls between 2 and 7. Substituting $x = 4$ into the first inequality, $2x + 3 > 7$, we get:

$2(4) + 3 > 7$

$8 + 3 > 7$

$11 > 7$

This is true, so $x = 4$ satisfies the first inequality. Now, let's substitute $x = 4$ into the second inequality, $x + 9 < 16$:

$4 + 9 < 16$

$13 < 16$

This is also true, so $x = 4$ satisfies the second inequality as well. Since $x = 4$ satisfies both inequalities, it confirms that our solution $2 < x < 7$ is likely correct. However, checking just one value is not a foolproof method. To be more certain, we can also test values outside the solution range to ensure they do not satisfy the inequalities. For instance, let's test $x = 1$ (which is less than 2) and $x = 8$ (which is greater than 7).

For $x = 1$:

$2(1) + 3 > 7$ becomes $5 > 7$, which is false.

$1 + 9 < 16$ becomes $10 < 16$, which is true. Since the first inequality is not satisfied, $x = 1$ is not a solution.

For $x = 8$:

$2(8) + 3 > 7$ becomes $19 > 7$, which is true.

$8 + 9 < 16$ becomes $17 < 16$, which is false. Since the second inequality is not satisfied, $x = 8$ is not a solution.

These checks further validate our solution $2 < x < 7$. The importance of checking solutions cannot be overstated. It helps prevent errors and reinforces understanding of the concepts involved. In more complex problems, checking solutions can be the key to identifying mistakes and ensuring the correctness of the final answer.

Real-World Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding and solving inequalities can help us make informed decisions and solve practical problems in everyday life. One common application is in budgeting and finance. For example, if you have a budget of $100 for groceries, you can use an inequality to represent the total cost of items you can purchase. If $x$ represents the cost of one item, the inequality might look like $nx ≤ 100$, where $n$ is the number of items. Solving this inequality helps you determine the maximum number of that item you can buy without exceeding your budget. In business and economics, inequalities are used to model supply and demand, profit margins, and cost constraints. For instance, a company might use inequalities to determine the range of prices at which they can sell a product to make a profit, considering production costs and market demand. Inequalities are also crucial in engineering and science. Engineers use inequalities to define tolerance levels for measurements, ensuring that components meet certain specifications. In physics, inequalities can represent the range of possible values for physical quantities, such as velocity or temperature. Computer science also relies heavily on inequalities. In algorithm analysis, inequalities are used to express the time complexity and space complexity of algorithms, helping developers optimize performance. Inequalities also play a role in cryptography, where they can be used to define security parameters and ensure the robustness of encryption methods.

Conclusion

Solving compound inequalities is a fundamental skill in mathematics with broad applications across various fields. In this article, we have walked through the step-by-step process of solving the compound inequality $2x + 3 > 7$ and $x + 9 < 16$. We began by understanding the concept of inequalities and compound inequalities, highlighting the importance of the connecting words “and” and “or.” We then solved each inequality separately, finding the solution sets $x > 2$ and $x < 7$. Combining these solutions using the “and” condition, we determined that the solution to the compound inequality is $2 < x < 7$. We also discussed how to represent this solution on a number line, providing a visual understanding of the range of values that satisfy the inequality. The importance of checking the solution was emphasized, and we demonstrated how to verify the answer by substituting values within and outside the solution range. Finally, we explored real-world applications of inequalities, showcasing their relevance in fields such as finance, business, engineering, and computer science. By mastering the techniques for solving compound inequalities, you gain a valuable tool for problem-solving in both academic and practical contexts. The ability to think critically about ranges and constraints is essential for making informed decisions and tackling complex challenges in various domains. Whether you are a student learning algebra or a professional applying mathematical concepts in your work, a solid understanding of inequalities will serve you well.