Solving Compound Inequality 2u - 2 ≥ -10 And 3u - 6 > -9

Jul 16, 2025 by ADMIN 57 views

Solve the Compound Inequality 2u - 2 ≥ -10 and 3u - 6 > -9: A Comprehensive Guide

Solving compound inequalities is a fundamental skill in algebra, often encountered in various mathematical contexts. This comprehensive guide will walk you through the step-by-step process of solving the compound inequality $2u - 2 ≥ -10$ and $3u - 6 > -9$ . We will cover each step in detail, ensuring a clear understanding of the underlying principles. Additionally, we will express the solution in interval notation, a standard format for representing solution sets, and address the possibility of an empty set solution, denoted by $∅$ . Whether you're a student tackling homework or someone looking to refresh your algebra skills, this guide provides a detailed explanation to help you master compound inequalities.

Understanding Compound Inequalities

Before diving into the solution, let's clarify what a compound inequality is. A compound inequality is a combination of two or more inequalities connected by the words "and" or "or." The word "and" indicates that both inequalities must be true simultaneously, while the word "or" indicates that at least one of the inequalities must be true. In our case, we have a compound inequality connected by "and," meaning we need to find the values of $u$ that satisfy both $2u - 2 ≥ -10$ and $3u - 6 > -9$ .

The Importance of "And" and "Or"

The distinction between "and" and "or" is crucial. An "and" compound inequality requires the solution to satisfy both inequalities. This means the solution set will be the intersection of the solution sets of the individual inequalities. Conversely, an "or" compound inequality requires the solution to satisfy at least one of the inequalities. The solution set will be the union of the solution sets of the individual inequalities. Understanding this difference is key to correctly solving compound inequalities and interpreting the results.

Visualizing Inequalities on a Number Line

A helpful way to visualize inequalities is by using a number line. For example, the inequality $x > 3$ represents all numbers greater than 3, which can be shown on a number line with an open circle at 3 and an arrow extending to the right. Similarly, $x ≤ 5$ represents all numbers less than or equal to 5, depicted with a closed circle at 5 and an arrow extending to the left. When solving compound inequalities, visualizing the solution sets on a number line can aid in determining the final solution, especially when dealing with "and" and "or" conditions. The intersection of two solution sets (for "and") will be the overlapping region, while the union of two solution sets (for "or") will include all regions covered by either inequality.

Step-by-Step Solution

Now, let's solve the given compound inequality: $2u - 2 ≥ -10$ and $3u - 6 > -9$ . We will tackle each inequality separately and then combine the results.

Solving the First Inequality: 2u - 2 ≥ -10

Isolate the term with u: To begin, we need to isolate the term containing $u$ . We can do this by adding 2 to both sides of the inequality:

$2u - 2 + 2 ≥ -10 + 2$

This simplifies to:

$2u ≥ -8$
Solve for u: Next, we solve for $u$ by dividing both sides of the inequality by 2:

$\frac{2u}{2} ≥ \frac{-8}{2}$

This gives us:

$u ≥ -4$

So, the solution to the first inequality is all values of $u$ greater than or equal to -4.

Solving the Second Inequality: 3u - 6 > -9

Isolate the term with u: Similar to the first inequality, we start by isolating the term with $u$ . Add 6 to both sides of the inequality:

$3u - 6 + 6 > -9 + 6$

This simplifies to:

$3u > -3$
Solve for u: Now, divide both sides of the inequality by 3:

$\frac{3u}{3} > \frac{-3}{3}$

This gives us:

$u > -1$

The solution to the second inequality is all values of $u$ greater than -1.

Combining the Solutions

We have found that $u ≥ -4$ and $u > -1$ . Since this is an "and" compound inequality, we need to find the values of $u$ that satisfy both conditions. In other words, we need the intersection of the two solution sets.

The first inequality, $u ≥ -4$ , includes all numbers from -4 to positive infinity, including -4. On a number line, this is represented by a closed circle at -4 and an arrow extending to the right.
The second inequality, $u > -1$ , includes all numbers greater than -1. On a number line, this is represented by an open circle at -1 and an arrow extending to the right.

The intersection of these two solution sets is the region where both conditions are met. Notice that any number greater than -1 is also greater than or equal to -4. Therefore, the solution to the compound inequality is $u > -1$ .

Expressing the Solution in Interval Notation

Interval notation is a concise way to represent sets of numbers. It uses parentheses and brackets to indicate whether the endpoints are included in the set.

A parenthesis ( or ) indicates that the endpoint is not included (open interval).
A bracket [ or ] indicates that the endpoint is included (closed interval).
Infinity $(∞)$ and negative infinity $(-∞)$ are always enclosed in parentheses because they are not specific numbers and cannot be included.

In our case, the solution is $u > -1$ . This means all numbers greater than -1 are included, but -1 itself is not. Therefore, in interval notation, the solution is $(-1, ∞)$ . The parenthesis next to -1 indicates that -1 is not included, and the parenthesis next to $∞$ indicates that infinity is not a specific endpoint.

Understanding Different Types of Intervals

To further clarify interval notation, let's look at some examples:

[a, b] represents all numbers between a and b, including a and b (closed interval).
(a, b) represents all numbers between a and b, excluding a and b (open interval).
[a, b) represents all numbers between a and b, including a but excluding b (half-open interval).
(a, b] represents all numbers between a and b, excluding a but including b (half-open interval).
(-∞, b] represents all numbers less than or equal to b.
(-∞, b) represents all numbers less than b.
[a, ∞) represents all numbers greater than or equal to a.
(a, ∞) represents all numbers greater than a.

Understanding these different notations is crucial for accurately representing solution sets.

No Solution: The Empty Set (∅)

In some cases, a compound inequality may have no solution. This occurs when the conditions of the inequalities contradict each other. For example, consider the compound inequality $x > 5$ and $x < 2$ . There is no number that can be simultaneously greater than 5 and less than 2. In such cases, the solution is the empty set, denoted by the symbol $∅$ .

Identifying Empty Set Solutions

To identify an empty set solution, it's important to analyze the individual inequalities and their solution sets. If the intersection of the solution sets is empty (i.e., there are no overlapping regions), then the compound inequality has no solution. Similarly, for an "or" compound inequality, if the union of the solution sets does not cover any real numbers, the solution is the empty set.

Example of an Empty Set Solution

Let's consider another example: $x ≤ -3$ and $x ≥ 4$ . The first inequality includes all numbers less than or equal to -3, while the second inequality includes all numbers greater than or equal to 4. There is no overlap between these two sets. Therefore, the solution to this compound inequality is the empty set, $∅$ .

Common Mistakes to Avoid

When solving compound inequalities, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them.

Misinterpreting "And" and "Or"

The most common mistake is confusing the "and" and "or" conditions. Remember, "and" requires both inequalities to be true, while "or" requires at least one to be true. Misinterpreting this can lead to incorrect solution sets.

Forgetting to Reverse the Inequality Sign

When multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign. Forgetting this rule is a frequent error that can significantly alter the solution.

Incorrectly Graphing on a Number Line

Visualizing the solution sets on a number line is a valuable tool, but incorrect graphing can lead to wrong answers. Ensure you use open circles for strict inequalities ( $<$ or $>$ ) and closed circles for inequalities that include equality $(≤$ or $≥$ ).

Not Expressing the Solution in Correct Interval Notation

Interval notation requires careful attention to detail. Using parentheses instead of brackets or vice versa can change the meaning of the solution set. Always double-check your interval notation to ensure it accurately reflects the solution.

Conclusion

In conclusion, solving the compound inequality $2u - 2 ≥ -10$ and $3u - 6 > -9$ involves several steps, including solving each inequality separately, combining the solutions based on the "and" condition, and expressing the final solution in interval notation. By following these steps carefully, we determined that the solution is $u > -1$ , which is represented in interval notation as $(-1, ∞)$ . Understanding compound inequalities, including the concepts of "and" and "or," is crucial for success in algebra and beyond. Remember to practice regularly and pay attention to details to avoid common mistakes. This comprehensive guide provides a solid foundation for mastering compound inequalities and confidently tackling similar problems in the future. Whether you are a student or someone reviewing algebraic concepts, this detailed explanation and step-by-step approach should enhance your understanding and problem-solving skills in mathematics. Remember, consistent practice and a clear understanding of the underlying principles are key to mastering mathematical concepts. By diligently applying these methods, you'll be well-equipped to solve a wide range of compound inequalities and other algebraic problems. Keep practicing, and you'll continue to improve your mathematical abilities.