Solving -16x > -80 A Step-by-Step Guide

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This article provides a detailed explanation of how to solve the inequality -16x > -80. We will explore the steps involved, the underlying principles, and why certain operations are performed. By the end of this guide, you will have a solid understanding of how to tackle similar inequalities and the logic behind the solution.

Understanding Inequalities

Before we dive into the specifics of this problem, let's briefly discuss what inequalities are and how they differ from equations. Inequalities, in mathematics, are expressions that use symbols like '>', '<', '≥', and '≤' to compare values. Unlike equations, which state that two expressions are equal, inequalities indicate a range of possible values. When we solve an inequality, we're finding the set of numbers that make the inequality true. The core idea in solving inequalities revolves around isolating the variable on one side, just like in equations, but with a crucial twist when multiplying or dividing by a negative number.

Step-by-Step Solution of -16x > -80

Now, let's break down the solution to the inequality -16x > -80 step by step. This will allow you to clearly understand the process and how each step changes the inequality.

1. The Initial Inequality

We start with the inequality:

-16x > -80

Our goal is to isolate 'x' on one side of the inequality. To do this, we need to get rid of the '-16' that is multiplying 'x'. This is a fundamental problem in algebra, and it is solved by performing the same operation on both sides of the equation.

2. Dividing Both Sides by -16

The key step here is to divide both sides of the inequality by -16. However, this is where the crucial difference between solving equations and inequalities comes into play. When we divide (or multiply) both sides of an inequality by a negative number, we must reverse the direction of the inequality sign. This is because multiplying or dividing by a negative number reflects the number line, changing the order of the numbers. For example, 2 < 3, but -2 > -3.

So, dividing both sides by -16, we get:

(-16x) / -16 < (-80) / -16

Notice that the '>' sign has changed to '<'.

This reversal of the inequality sign is a critical concept in solving inequalities, and failure to do so will lead to an incorrect solution. Think of it this way: multiplying or dividing by a negative flips the number line, so you need to flip the inequality to maintain the correct relationship.

3. Simplifying the Inequality

Now, let's simplify both sides of the inequality:

x < 5

On the left side, -16x divided by -16 simplifies to x. On the right side, -80 divided by -16 equals 5. Thus, we arrive at the solution: x < 5.

This means that any value of x that is less than 5 will satisfy the original inequality. For instance, if we substitute x = 0 into the original inequality, we get -16(0) > -80, which simplifies to 0 > -80, which is true. If we substitute x = 6 (which is not less than 5), we get -16(6) > -80, which simplifies to -96 > -80, which is false.

4. Understanding the Solution Set

The solution x < 5 represents a set of numbers. Specifically, it represents all real numbers less than 5. This can be visualized on a number line as an open interval extending from negative infinity up to, but not including, 5. The open circle at 5 indicates that 5 is not included in the solution set.

Why Does the Inequality Sign Flip?

It’s essential to understand why we flip the inequality sign when multiplying or dividing by a negative number. Let’s illustrate this with a simple example:

Consider the inequality:

2 < 4

This is clearly true. Now, let’s multiply both sides by -1:

-2 -4

If we kept the inequality sign the same, we would have -2 < -4, which is false. However, if we flip the inequality sign, we get -2 > -4, which is true. This demonstrates why the sign must be flipped to maintain the truth of the inequality.

This flipping behavior can also be explained by thinking about the number line. Multiplying by a negative number essentially reflects the numbers across the origin (0). This reflection reverses the order of the numbers, hence the need to reverse the inequality sign.

Common Mistakes to Avoid

When solving inequalities, it’s easy to make mistakes if you’re not careful. Here are some common pitfalls to watch out for:

  1. Forgetting to Flip the Sign: The most common mistake is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Always double-check this step!

  2. Incorrectly Distributing Negatives: If there are parentheses in the inequality, make sure to distribute negative signs correctly. A mistake here can completely change the outcome.

  3. Combining Terms Incorrectly: Be sure to combine like terms on each side of the inequality before isolating the variable. Mixing up terms can lead to an incorrect solution.

  4. Misinterpreting the Solution Set: Understand what your solution means. For example, x < 5 includes all numbers less than 5, but not 5 itself. Pay attention to whether the inequality is strict (>, <) or inclusive (≥, ≤).

Alternative Methods for Solving the Inequality

While dividing by -16 is the most direct approach, there are alternative ways to solve the inequality. One method is to add 16x and 80 to both sides:

-16x > -80

Add 16x to both sides:

0 > -80 + 16x

Add 80 to both sides:

80 > 16x

Now, divide both sides by 16 (since 16 is positive, we don't flip the sign):

80 / 16 > x

5 > x

This is the same as x < 5.

This alternative method avoids dividing by a negative number, which can be helpful for students who are prone to forgetting to flip the sign. It reinforces the idea that there are often multiple paths to arrive at the correct solution.

Practice Problems

To solidify your understanding, let’s work through a few practice problems:

  1. Solve: -3x + 5 < 14

  2. Solve: 2(x - 1) ≥ -6

  3. Solve: -5x - 10 > 25

Solving these problems independently will help reinforce the concepts discussed and build confidence in your ability to tackle inequalities.

Real-World Applications of Inequalities

Inequalities aren't just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

  1. Budgeting: Inequalities are used to set budget constraints. For example, if you have $100 to spend, you can represent this as an inequality: spending ≤ $100.

  2. Speed Limits: Speed limits are expressed as inequalities. For example, the speed limit on a highway might be speed ≤ 65 mph.

  3. Temperature Ranges: Weather forecasts often use inequalities to express temperature ranges. For instance, the forecast might say the temperature will be between 70°F and 80°F, which can be written as 70 ≤ temperature ≤ 80.

  4. Manufacturing: Inequalities are used to set tolerances in manufacturing. For example, the diameter of a bolt might need to be within a certain range: diameter ± 0.1 mm.

Understanding these applications helps to see the practical relevance of solving inequalities and reinforces the importance of mastering this skill.

Conclusion

Solving inequalities is a fundamental skill in algebra and has wide-ranging applications. By understanding the principles behind each step, particularly the rule about flipping the inequality sign when multiplying or dividing by a negative number, you can confidently solve a wide variety of problems. Remember to practice regularly and be mindful of common mistakes. With consistent effort, you'll become proficient in solving inequalities.

In summary, the solution to the inequality -16x > -80 is x < 5. This means that any number less than 5 will satisfy the original inequality. This detailed explanation should provide you with a solid understanding of how to solve this type of inequality and the reasoning behind each step.