Solving Inequalities Finding Solutions For X

Jul 9, 2025 by Admin 45 views

Solving Inequalities A Step-by-Step Guide to Finding the Solution Set

In this article, we will delve into the process of solving inequalities, providing a step-by-step guide to finding the solution set for the given compound inequality: 3x - 91 > -87 OR 21x - 17 > 25. Inequalities, unlike equations, involve a range of possible solutions rather than a single value. Understanding how to manipulate and solve them is crucial in various fields, including mathematics, physics, and economics.

Understanding Inequalities

Before we dive into the solution, let's first understand the basics of inequalities. Inequalities are mathematical statements that compare two expressions using symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Solving an inequality means finding the set of values that satisfy the given condition. This set of values is known as the solution set.

In our case, we have a compound inequality, which consists of two inequalities connected by the word "OR." This means that the solution set includes all values of x that satisfy either the first inequality, the second inequality, or both. To solve this, we'll solve each inequality separately and then combine their solutions.

Solving the First Inequality: 3x - 91 > -87

To isolate x, we need to perform a series of algebraic operations on both sides of the inequality. Our goal is to get x by itself on one side of the inequality sign.

Add 91 to both sides: This eliminates the -91 on the left side.
```
3x - 91 + 91 > -87 + 91
```
This simplifies to:
```
3x > 4
```
Divide both sides by 3: This isolates x on the left side.
```
3x / 3 > 4 / 3
```
This gives us the solution for the first inequality:
```
x > 4/3
```

So, the solution to the first inequality is all values of x greater than 4/3. This means any number larger than 4/3 will make the inequality 3x - 91 > -87 true. For example, if we substitute x = 2 (which is greater than 4/3) into the inequality, we get:

3(2) - 91 > -87
6 - 91 > -87
-85 > -87

This is a true statement, confirming that x = 2 is indeed part of the solution set for the first inequality. Similarly, any other value greater than 4/3 will satisfy this inequality.

Solving the Second Inequality: 21x - 17 > 25

Now, let's solve the second inequality, 21x - 17 > 25, using a similar approach to isolate x.

Add 17 to both sides: This eliminates the -17 on the left side.
```
21x - 17 + 17 > 25 + 17
```
This simplifies to:
```
21x > 42
```
Divide both sides by 21: This isolates x on the left side.
```
21x / 21 > 42 / 21
```
This gives us the solution for the second inequality:
```
x > 2
```

Thus, the solution to the second inequality is all values of x greater than 2. This means any number larger than 2 will satisfy the inequality 21x - 17 > 25. For instance, if we substitute x = 3 (which is greater than 2) into the inequality, we get:

21(3) - 17 > 25
63 - 17 > 25
46 > 25

This is a true statement, indicating that x = 3 is part of the solution set for the second inequality. Any other value greater than 2 will also satisfy this inequality.

Combining the Solutions

Since our original problem is a compound inequality connected by "OR," we need to combine the solutions of both inequalities. The solution set for the compound inequality includes all values of x that satisfy either x > 4/3 OR x > 2.

Notice that if x > 2, then x is also greater than 4/3 (since 2 is greater than 4/3). Therefore, the solution x > 2 encompasses all values that satisfy x > 4/3. In other words, if a number is greater than 2, it is automatically greater than 4/3. This means we don't need to consider x > 4/3 separately because it's already included in the solution x > 2.

To visualize this, consider a number line. The solution x > 4/3 includes all points to the right of 4/3, while the solution x > 2 includes all points to the right of 2. Since 2 is to the right of 4/3 on the number line, all points to the right of 2 are also to the right of 4/3. Thus, the combined solution is simply x > 2.

Therefore, the solution to the compound inequality 3x - 91 > -87 OR 21x - 17 > 25 is x > 2. This means that any value of x greater than 2 will satisfy the original compound inequality. For example, if we substitute x = 3 into both original inequalities, we get:

3(3) - 91 > -87  OR  21(3) - 17 > 25
9 - 91 > -87  OR  63 - 17 > 25
-82 > -87  OR  46 > 25

Both of these inequalities are true, confirming that x = 3 is a solution. Similarly, any value greater than 2 will satisfy at least one of the inequalities, making it a solution to the compound inequality.

Final Answer

Therefore, the solution to the compound inequality 3x - 91 > -87 OR 21x - 17 > 25 is:

(A) x > 2

This comprehensive guide has walked you through the process of solving compound inequalities. By understanding the fundamental principles and applying the step-by-step approach, you can confidently tackle similar problems in the future. Remember to always isolate the variable and consider the logical connection (OR, AND) between the inequalities when combining the solutions.

Key Concepts Recap

Inequalities: Mathematical statements comparing expressions using >, <, ≥, or ≤.
Compound Inequalities: Two or more inequalities connected by "OR" or "AND."
Solution Set: The set of values that satisfy the inequality or inequalities.
Isolating the Variable: The process of performing algebraic operations to get the variable by itself on one side of the inequality.
Combining Solutions: For "OR" inequalities, the solution set includes values that satisfy either inequality; for "AND" inequalities, the solution set includes values that satisfy both inequalities.

By mastering these concepts and practicing regularly, you'll develop the skills necessary to solve a wide range of inequality problems.