Determining Solutions For The Inequality 9 Less Than 2 * Or - 5
In this article, we will explore how to determine whether given values of 'or' satisfy the inequality 9 < 2 * or - 5. This involves substituting each value into the inequality and checking if the resulting statement is true. We'll walk through the process step-by-step for each value provided, ensuring a clear understanding of how to solve such problems. Understanding inequalities is crucial in mathematics as they help us define ranges and conditions for variables.
Understanding Inequalities
Before we dive into the specific problem, let's first understand what inequalities are and how they work. Inequalities are mathematical expressions that compare two values, showing that one value is greater than, less than, greater than or equal to, or less than or equal to another value. The symbols used in inequalities include:
- < (less than)
- > (greater than)
- ≤ (less than or equal to)
- ≥ (greater than or equal to)
Solving inequalities involves finding the range of values that make the inequality true. This is a fundamental concept in algebra and is used extensively in various mathematical applications. In our case, we have the inequality 9 < 2 * or - 5, and we want to determine which given values of 'or' satisfy this condition.
Method for Determining Solutions
To determine if a given value of 'or' is a solution to the inequality, we follow a simple process:
- Substitute: Replace the variable 'or' in the inequality with the given value.
- Simplify: Perform the necessary arithmetic operations to simplify the expression.
- Evaluate: Check if the resulting statement is true. If it is true, then the value is a solution; otherwise, it is not.
We will apply this method to each value provided in the table to determine whether it is a solution to the inequality 9 < 2 * or - 5.
Analyzing the Values of 'or'
Case 1: or = 7
Let's start by substituting or = 7 into the inequality:
9 < 2 * (7) - 5
Now, simplify the expression:
9 < 14 - 5
9 < 9
This statement is false because 9 is not less than 9. Therefore, 7 is not a solution to the inequality. In this case, the equality 9 = 9 holds, but the inequality 9 < 9 does not. This distinction is crucial when working with inequalities, as we are looking for values that make the statement strictly less than, in this context.
Case 2: or = -9
Next, we substitute or = -9 into the inequality:
9 < 2 * (-9) - 5
Simplify the expression:
9 < -18 - 5
9 < -23
This statement is also false because 9 is not less than -23. Therefore, -9 is not a solution to the inequality. The result clearly shows that substituting -9 for 'or' leads to a contradiction, further illustrating that negative values for 'or' do not satisfy the given inequality.
Case 3: or = -4
Now, let's substitute or = -4 into the inequality:
9 < 2 * (-4) - 5
Simplify the expression:
9 < -8 - 5
9 < -13
This statement is false because 9 is not less than -13. Therefore, -4 is not a solution to the inequality. Similar to the previous case, substituting a negative value for 'or' does not result in a true statement, reinforcing the understanding that negative values are not suitable solutions for this inequality.
Case 4: or = 8
Finally, we substitute or = 8 into the inequality:
9 < 2 * (8) - 5
Simplify the expression:
9 < 16 - 5
9 < 11
This statement is true because 9 is less than 11. Therefore, 8 is a solution to the inequality. This case demonstrates that positive values, specifically 8, can indeed satisfy the inequality, highlighting the importance of testing different types of values to fully understand the solution set.
Summary of Solutions
| or | Is it a solution? | Explanation |
|---|---|---|
| 7 | No | Substituting 7 into the inequality results in 9 < 9, which is false. |
| -9 | No | Substituting -9 into the inequality results in 9 < -23, which is false. |
| -4 | No | Substituting -4 into the inequality results in 9 < -13, which is false. |
| 8 | Yes | Substituting 8 into the inequality results in 9 < 11, which is true. |
Conclusion
In conclusion, we have determined which values of 'or' are solutions to the inequality 9 < 2 * or - 5. By substituting each value and simplifying the expression, we found that only or = 8 satisfies the inequality. The other values, 7, -9, and -4, do not make the inequality true. This exercise illustrates the process of solving inequalities and the importance of careful substitution and evaluation. Understanding inequalities is crucial for various mathematical applications, and this step-by-step approach provides a solid foundation for tackling more complex problems. This detailed analysis not only answers the question but also reinforces the fundamental concepts of inequality solutions, offering a comprehensive understanding for anyone studying algebra.
