Solving The Inequality $16 < 5w - 9$ A Step-by-Step Guide

Understanding and solving inequalities is a fundamental concept in mathematics. Inequalities, unlike equations, deal with relationships where one value is greater than, less than, or not equal to another. This article will provide a detailed, step-by-step guide to solving the inequality $16 < 5w - 9$, ensuring clarity and comprehension for anyone looking to master this essential skill. Whether you're a student tackling algebra or someone brushing up on math skills, this guide will break down each step with clear explanations and helpful tips.

Understanding Inequalities

Before diving into the solution, it's crucial to grasp the basics of inequalities. An inequality is a mathematical statement that compares two expressions using symbols such as:

• < (less than)

• (greater than)

• ≤ (less than or equal to)
• ≥ (greater than or equal to)

The inequality $16 < 5w - 9$ states that 16 is less than the expression $5w - 9$. Our goal is to find the values of $w$ that satisfy this condition. Solving inequalities involves similar steps to solving equations, but with one key difference: multiplying or dividing by a negative number reverses the inequality sign. This concept is vital to remember to avoid common mistakes. Let's explore the step-by-step process of solving our given inequality, ensuring we understand each operation and its impact on the solution.

Step 1 Isolate the Term with the Variable

The first step in solving the inequality $16 < 5w - 9$ is to isolate the term containing the variable, which in this case is $5w$. To do this, we need to eliminate the constant term, $-9$, from the right side of the inequality. We can accomplish this by adding 9 to both sides of the inequality. This maintains the balance of the inequality, similar to how adding the same value to both sides of an equation keeps it balanced. Adding 9 to both sides gives us:

$16 + 9 < 5w - 9 + 9$

Simplifying this, we get:

$25 < 5w$

Now, we have successfully isolated the term with the variable on one side of the inequality. This step is crucial because it brings us closer to isolating the variable itself. By performing the same operation on both sides, we ensure that the inequality remains valid and the solution set remains unchanged. Understanding this foundational principle is key to tackling more complex inequalities in the future. The next step involves isolating the variable $w$ completely, which we will cover in the subsequent section.

Step 2 Isolate the Variable

Having isolated the term $5w$, our next step is to isolate the variable $w$ itself. Currently, $w$ is being multiplied by 5. To undo this multiplication, we need to perform the inverse operation, which is division. We will divide both sides of the inequality by 5. Since 5 is a positive number, we do not need to reverse the inequality sign. Dividing both sides by 5, we get:

rac{25}{5} < rac{5w}{5}

Simplifying this, we find:

$5 < w$

This inequality tells us that 5 is less than $w$, which can also be read as $w$ is greater than 5. This is the solution to our inequality. By dividing both sides by the coefficient of $w$, we have successfully isolated the variable and determined the range of values that satisfy the original inequality. It’s important to remember that if we were dividing by a negative number, we would need to flip the inequality sign. Now that we have our solution, let’s explore how to interpret and represent it.

Step 3 Interpret the Solution

The solution $5 < w$ (or equivalently, $w > 5$) means that any value of $w$ greater than 5 will satisfy the original inequality $16 < 5w - 9$. It's crucial to understand what this solution represents. In simpler terms, $w$ can be any number larger than 5, but not including 5 itself. For instance, 5.01, 6, 10, or 100 are all valid solutions because they are greater than 5.

To solidify your understanding, let's test a couple of values. First, let's try $w = 6$, which is greater than 5. Plugging this into the original inequality:

$16 < 5(6) - 9$

$16 < 30 - 9$

$16 < 21$

This statement is true, confirming that $w = 6$ is indeed a solution. Now, let's test a value that is not greater than 5, such as $w = 5$:

$16 < 5(5) - 9$

$16 < 25 - 9$

$16 < 16$

This statement is false because 16 is not less than 16. This confirms that $w$ must be strictly greater than 5. Understanding how to interpret and verify solutions is an essential skill in algebra and beyond. Now, let's discuss how to represent this solution graphically.

Representing the Solution on a Number Line

Visualizing the solution on a number line is a powerful way to understand the range of values that satisfy the inequality. To represent $w > 5$ on a number line, we first draw a number line and mark the point 5. Since $w$ is strictly greater than 5, we use an open circle (or a parenthesis) at 5 to indicate that 5 itself is not included in the solution. Then, we draw an arrow extending to the right from 5, indicating that all values greater than 5 are part of the solution.

Here’s a visual representation:

<----------------|--------------------)-------------------->
                4                    5                    6

In this number line:

• The open circle at 5 indicates that 5 is not included.
• The arrow extending to the right indicates that all values greater than 5 are included.

This graphical representation provides a clear visual understanding of the solution set. It helps to reinforce the concept that inequalities represent a range of values rather than a single value, as in equations. Understanding how to represent solutions on a number line is a valuable skill for visualizing and interpreting inequalities. Let's now address some common mistakes to avoid when solving inequalities.

Common Mistakes to Avoid

When solving inequalities, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them and ensure accuracy in your problem-solving. Here are some key mistakes to watch out for:

1. Forgetting to Flip the Inequality Sign: The most critical mistake is failing to reverse the inequality sign when multiplying or dividing both sides by a negative number. For example, if you have $-2w > 6$, dividing by -2 requires you to flip the sign to get $w < -3$. Always double-check this step when dealing with negative coefficients.
2. Incorrectly Distributing: When an inequality involves parentheses, ensure you distribute correctly. For instance, in $3(w + 2) < 9$, distribute the 3 to both terms inside the parentheses: $3w + 6 < 9$. A common error is to only multiply one term, leading to an incorrect simplification.
3. Adding or Subtracting Incorrectly: Simple arithmetic errors can lead to wrong answers. Ensure you accurately add or subtract terms when isolating the variable. Double-checking your arithmetic can save you from these mistakes.
4. Misinterpreting the Inequality Sign: It’s crucial to understand the meaning of each inequality sign. For example, $w > 5$ means $w$ is greater than 5, not greater than or equal to 5. Using the wrong sign in your final answer can change the solution set entirely.
5. Not Checking the Solution: Always verify your solution by plugging a value from your solution set back into the original inequality. This helps you confirm that your answer is correct and identify any potential errors in your steps.

By being mindful of these common mistakes and practicing regularly, you can improve your accuracy and confidence in solving inequalities. Now, let's summarize the steps we've covered in solving the inequality $16 < 5w - 9$.

Summary of Steps

To recap, solving the inequality $16 < 5w - 9$ involves the following steps:

1. Isolate the Term with the Variable: Add 9 to both sides of the inequality to get $25 < 5w$.
2. Isolate the Variable: Divide both sides by 5 to get $5 < w$, which is equivalent to $w > 5$.
3. Interpret the Solution: Understand that $w > 5$ means $w$ can be any value greater than 5.
4. Represent the Solution on a Number Line: Use an open circle at 5 and an arrow extending to the right to visualize the solution.
5. Check the Solution: Substitute a value greater than 5 into the original inequality to verify the solution.

By following these steps carefully, you can confidently solve a wide range of inequalities. Understanding and mastering these techniques is crucial for success in algebra and higher-level mathematics. Now, let's conclude with some final thoughts on the importance of practicing and applying these skills.

Conclusion

Solving inequalities is a foundational skill in mathematics with applications in various fields, from science and engineering to economics and finance. By understanding the basic principles and practicing regularly, you can develop the proficiency needed to tackle more complex problems. Remember the key steps: isolate the variable term, isolate the variable itself, interpret the solution, and represent it visually. Also, be mindful of common mistakes, especially the rule about flipping the inequality sign when multiplying or dividing by a negative number.

The inequality $16 < 5w - 9$ serves as a great example to illustrate these steps. By breaking down the problem into manageable parts and understanding the logic behind each operation, you can confidently arrive at the solution $w > 5$. Keep practicing with different inequalities to reinforce your understanding and build your problem-solving skills. With consistent effort, you'll find that solving inequalities becomes second nature, opening up new avenues for mathematical exploration and application.