Solving The Inequality -2(x-5) < 10 A Step-by-Step Guide

by THE IDEN 57 views

In the realm of mathematics, solving inequalities is a fundamental skill. This article delves into the solution of the inequality -2(x-5) < 10, within the set of integers from -2 to 2. We will embark on a step-by-step journey, meticulously unpacking the inequality and arriving at the correct solution. Our focus is to provide a clear, concise, and comprehensive explanation that caters to both beginners and those seeking a refresher.

We are tasked with finding the integer solutions to the inequality -2(x-5) < 10 within the set $ {x \mid -2 \leq x \leq 2, x \in \mathbb{Z}} $. This means we need to identify which integers between -2 and 2 (inclusive) satisfy the given inequality. The options presented are:

  1. 0, 1, 2
  2. 1, 2
  3. -2, -1, 0
  4. -2, -1

To solve the inequality -2(x-5) < 10, we need to isolate x on one side. We'll achieve this through a series of algebraic manipulations. Let's break down the process:

1. Distribute the -2

The first step is to distribute the -2 across the terms inside the parentheses:

-2(x-5) < 10 becomes -2x + 10 < 10

This step involves applying the distributive property of multiplication over subtraction. We multiply -2 by both x and -5, remembering that a negative times a negative results in a positive.

2. Subtract 10 from both sides

Next, we subtract 10 from both sides of the inequality to isolate the term with x:

-2x + 10 - 10 < 10 - 10 simplifies to -2x < 0

This step maintains the inequality because subtracting the same value from both sides doesn't change the relationship between them.

3. Divide both sides by -2

Now, we divide both sides by -2 to isolate x. Crucially, when we divide or multiply an inequality by a negative number, we must reverse the inequality sign:

-2x / -2 > 0 / -2 becomes x > 0

This is a critical step. Many errors in solving inequalities arise from forgetting to flip the inequality sign when dividing by a negative number. The inequality sign flips because we are essentially multiplying both sides by -1/2, which changes the direction of the inequality.

4. Consider the given set

We've determined that x must be greater than 0. However, we also have the constraint that x must be an integer within the set {-2, -1, 0, 1, 2}. We now need to identify the integers within this set that satisfy x > 0.

5. Identify the Solutions

The integers in the set {-2, -1, 0, 1, 2} that are greater than 0 are 1 and 2. Therefore, the solution set is {1, 2}.

Now, let's compare our solution set with the given options:

  1. 0, 1, 2 - This option includes 0, which does not satisfy x > 0.
  2. 1, 2 - This option matches our solution set.
  3. -2, -1, 0 - This option includes negative numbers and 0, none of which satisfy x > 0.
  4. -2, -1 - This option includes only negative numbers, which do not satisfy x > 0.

Therefore, the correct answer is option 2: 1, 2.

Solving inequalities involves a few potential pitfalls. Being aware of these can help you avoid errors:

  • Forgetting to Flip the Inequality Sign: This is the most common mistake. Always remember to reverse the inequality sign when multiplying or dividing by a negative number.
  • Incorrect Distribution: Ensure you distribute the negative sign correctly when dealing with parentheses. For example, -2(x-5) should be -2x + 10, not -2x - 10.
  • Ignoring the Given Set: Always consider the constraints of the problem. In this case, we were limited to integers between -2 and 2. Failing to consider this could lead to an incorrect answer.
  • Misinterpreting the Inequality Sign: Make sure you understand what the inequality sign means. For example, x > 0 means x is greater than 0, not greater than or equal to 0.

While we've solved this inequality algebraically, it's worth noting that you could also approach it by testing each integer within the given set. This can be a useful strategy, especially for simpler inequalities or when you're unsure about the algebraic steps. However, for more complex inequalities, the algebraic method is generally more efficient and less prone to error.

Inequalities are not just abstract mathematical concepts; they have numerous real-world applications. They are used in:

  • Optimization Problems: Inequalities help define constraints in optimization problems, where the goal is to find the best solution within certain limits.
  • Statistics: Inequalities are used in hypothesis testing and confidence interval calculations.
  • Economics: Inequalities are used to model supply and demand, and to analyze economic growth.
  • Computer Science: Inequalities are used in algorithm design and analysis.
  • Everyday Life: Inequalities are used in budgeting, time management, and decision-making.

For instance, consider a scenario where you have a budget of $100 for groceries. If x represents the amount you spend, you can express your budget constraint as x ≤ 100. This inequality helps you make informed decisions while grocery shopping.

In this article, we have thoroughly explored the solution to the inequality -2(x-5) < 10 within the set of integers from -2 to 2. We meticulously walked through each step, highlighting key concepts and potential pitfalls. By understanding the principles of solving inequalities, you can confidently tackle similar problems and apply these skills to various real-world scenarios. Remember to always double-check your work, especially when dealing with negative numbers and inequality signs. The solution to the inequality, considering the given set, is 1, 2.

To enhance the search engine optimization (SEO) of this article, we can incorporate relevant keywords strategically. These keywords will help users find this article when searching for information on solving inequalities and related topics. Here are some of the keywords we can use:

  • Solving inequalities
  • Algebraic inequalities
  • Integer solutions
  • Inequality problem
  • Step-by-step solution
  • Distributive property
  • Flipping inequality sign
  • Real-world applications
  • Mathematical inequalities
  • Problem-solving

By incorporating these keywords naturally within the article's content, headings, and meta descriptions, we can improve its visibility in search engine results.

To solidify your understanding of solving inequalities, consider practicing with additional problems. You can find practice problems in textbooks, online resources, or by creating your own variations of the problem we solved in this article. The key to mastering this skill is consistent practice and a clear understanding of the underlying principles.

Solving inequalities is a fundamental skill in mathematics with wide-ranging applications. By understanding the steps involved and being mindful of potential pitfalls, you can confidently solve a variety of inequality problems. Remember to practice regularly and seek clarification when needed. With dedication and a solid understanding of the concepts, you can master this essential mathematical skill.