Solving Inequalities: Find The Correct One For X = -18
Hey math enthusiasts! Today, we're diving into the world of inequalities. Specifically, we're going to figure out which inequality holds true when the value of x is -18. This is a fundamental concept in algebra, so understanding it well will set you up for success in more complex math problems. Let's break down the problem step-by-step and uncover the correct answer together.
Understanding the Basics of Inequalities
Before we jump into the specific inequalities, let's refresh our memory on the basics. An inequality is a mathematical statement that compares two values, indicating that they are not equal. Instead of an equals sign (=), inequalities use symbols such as:
- Less than (<)
- Greater than (>)
- Less than or equal to (≤)
- Greater than or equal to (≥)
Solving an inequality is similar to solving an equation. We aim to isolate the variable (in our case, x) on one side of the inequality. However, there's a crucial rule to remember: When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is a common point where mistakes can happen, so pay close attention!
For example, if we have the inequality x > 5 and multiply both sides by -1, we get -x < -5. The inequality sign flips from greater than to less than.
Now, let's look at the given options and substitute x = -18 into each one to determine which inequality is true. Remember, the goal is to find the inequality that results in a true statement after we substitute the value of x.
Let’s explore the importance of inequalities. Inequalities play a huge role in various mathematical and real-world scenarios. Inequalities are used to express constraints, represent ranges of values, and describe relationships between quantities. For instance, in economics, inequalities can model budget constraints or demand curves. In physics, they help define the limits of certain physical quantities, such as energy or momentum. Moreover, inequalities are crucial in computer science, specifically in algorithm design and optimization, for defining boundaries and checking conditions. Understanding inequalities not only helps in solving mathematical problems but also provides a framework for analyzing and interpreting real-world situations involving limits, comparisons, and constraints. This ability to model and solve inequalities is critical across numerous fields, making it an essential skill to master.
Furthermore, inequalities are foundational for more advanced mathematical concepts like calculus and linear programming. In calculus, inequalities are used to define intervals of convergence for series and integrals. In linear programming, inequalities are used to define the feasible region of optimization problems, where the goal is to maximize or minimize a function subject to certain constraints. Mastering inequalities therefore provides a strong base for tackling more complex mathematical ideas. The application of inequalities spans across many disciplines, underlining their versatility and importance in both theoretical and practical contexts. This makes the understanding of inequalities extremely essential in various fields and reinforces the need for a thorough understanding of these concepts.
Analyzing the Inequalities
Let's meticulously analyze each of the provided inequalities by substituting x = -18 and simplifying the expressions. This step is crucial for determining which inequality is true. Remember to pay close attention to the arithmetic operations, especially the handling of negative signs. Accuracy in these calculations is paramount to arrive at the correct answer. Let's take each option one at a time.
A. -x - 6.5 ≤ 9
Substitute x = -18:
-(-18) - 6.5 ≤ 9
Simplify:
18 - 6.5 ≤ 9
11.5 ≤ 9
This statement is false because 11.5 is not less than or equal to 9.
B. x - 6.5 ≥ -9
Substitute x = -18:
-18 - 6.5 ≥ -9
Simplify:
-24.5 ≥ -9
This statement is also false because -24.5 is not greater than or equal to -9.
C. x - 6.5 ≤ -9
Substitute x = -18:
-18 - 6.5 ≤ -9
Simplify:
-24.5 ≤ -9
This statement is true because -24.5 is less than or equal to -9.
D. -x - 6.5 ≤ -9
Substitute x = -18:
-(-18) - 6.5 ≤ -9
Simplify:
18 - 6.5 ≤ -9
11.5 ≤ -9
This statement is false because 11.5 is not less than or equal to -9.
Through this process, we have verified each inequality by substituting the given value of x and simplifying the expressions. This method, while straightforward, is very effective in ensuring accuracy. It is a fundamental technique for checking the validity of inequalities under specific conditions. Remember, in algebra, precision in calculations and a strong understanding of inequality rules are essential for successful problem-solving. This exercise showcases how a systematic approach can help one confidently determine the correct solution, while also understanding the incorrect options.
The ability to critically assess inequalities, coupled with an understanding of basic arithmetic, builds a solid foundation for more advanced mathematical concepts. This practice not only reinforces fundamental skills but also fosters a logical approach to problem-solving, which is critical in various areas beyond mathematics. For example, it can be applied to logical reasoning in other subjects and even in everyday decision-making, where one must evaluate different options based on given constraints and conditions. The skills gained from solving inequalities, such as analyzing, substituting, and simplifying, are invaluable. Therefore, practicing and mastering these techniques is not only beneficial for mathematical proficiency but also promotes general cognitive development.
Determining the Correct Answer
After substituting x = -18 into each inequality and simplifying, we found that only one statement is true. Option C, which is x - 6.5 ≤ -9, holds true when x is -18. The other options resulted in false statements. Therefore, the correct answer is C.
This process highlights the significance of substituting values correctly and carefully. It shows how precision in arithmetic and a solid understanding of inequality symbols can lead you to the right answer. Always double-check your calculations, especially when dealing with negative numbers. This systematic approach is key to success in algebra and beyond.
Now, let's explore some key takeaways from this exercise. We learned the importance of understanding inequality symbols, the rules for manipulating inequalities, and how to substitute values to test the validity of an inequality. Remember these key points as you tackle more complex math problems. Practice is the key to mastering inequalities. The more you practice, the more comfortable and confident you'll become in solving these types of problems.
Here’s a quick recap: Substituting the value of x into each inequality and simplifying allowed us to evaluate the truthfulness of each statement. The correct option was the one that produced a true statement after the substitution and simplification. Understanding how to handle negative numbers and inequality signs is vital to achieve accuracy in algebraic calculations. This method can be applied to solve various other algebraic problems, emphasizing the importance of a step-by-step approach and careful attention to detail. This method underscores how critical a clear, logical thought process is in mathematics.
Conclusion: Mastering Inequalities
Congratulations! You've successfully navigated through this inequality problem. Remember, the core of solving such problems involves carefully substituting values, performing accurate calculations, and understanding the rules of inequalities. Keep practicing, and you'll become more confident in your ability to solve these types of problems. Inequalities are a fundamental concept in mathematics. The principles learned here are applicable to more advanced mathematical topics and real-world scenarios. A strong grasp of inequalities lays a good foundation for tackling a wide range of mathematical and practical problems. Keep exploring, keep learning, and enjoy the journey of mastering mathematics!
For further study, you can explore more resources related to inequalities, such as:
- Khan Academy: A great platform for learning various mathematical concepts, including inequalities.
- Math is Fun: Offers clear explanations and examples of inequalities.
Here's an external link for more info
This website provides detailed lessons and examples to help you understand and master inequalities. Happy learning!
