Solving Inequalities A Step-by-Step Guide To 58-10x ≤ 20+9x

Understanding Linear Inequalities

When dealing with linear inequalities, our goal is to isolate the variable on one side of the inequality symbol, much like solving linear equations. However, there's a crucial difference: when we multiply or divide both sides of an inequality by a negative number, we must flip the direction of the inequality sign. This is because multiplying or dividing by a negative value reverses the order of the numbers on the number line. In this comprehensive guide, we'll walk through the step-by-step process of solving the inequality $58 - 10x ext{ ≤ } 20 + 9x$, explaining each operation and the reasoning behind it. This exploration will not only provide a solution but also enhance your understanding of handling inequalities in various mathematical contexts. Grasping the nuances of linear inequalities is crucial for success in algebra and beyond, where these concepts form the foundation for more complex problem-solving techniques. Through this detailed explanation, you’ll gain confidence in your ability to manipulate and solve inequalities accurately, ensuring you are well-prepared for future mathematical challenges. The ability to solve inequalities is fundamental in many areas of mathematics, including calculus, optimization, and mathematical modeling. Mastering these skills opens doors to a deeper understanding of mathematical principles and their applications in real-world scenarios.

Step-by-Step Solution

To solve the inequality $58 - 10x ext{ ≤ } 20 + 9x$, we need to isolate the variable x. Let's go through the steps:

1. Combine Like Terms

The first step is to gather all terms containing x on one side of the inequality and constant terms on the other side. We can start by adding $10x$ to both sides of the inequality. This eliminates the $-10x$ term on the left side and moves the x terms to the right side. This is a common strategy in solving inequalities and equations, aimed at simplifying the expression and making it easier to isolate the variable. By doing so, we are essentially rearranging the inequality without changing its solution set. Adding the same term to both sides maintains the balance and ensures the inequality remains valid.

$58 - 10x + 10x ext{ ≤ } 20 + 9x + 10x$

Simplifies to:

$58 ext{ ≤ } 20 + 19x$

Next, we want to isolate the term with x. To do this, subtract 20 from both sides of the inequality. This step ensures that we move all constant terms to one side, further simplifying the inequality. Subtracting the same value from both sides maintains the balance of the inequality, keeping the solution set unchanged. This operation is crucial for isolating the variable and moving closer to the final solution.

$58 - 20 ext{ ≤ } 20 + 19x - 20$

Which simplifies to:

$38 ext{ ≤ } 19x$

2. Isolate the Variable

Now, to completely isolate x, we need to divide both sides of the inequality by 19. Since 19 is a positive number, we do not need to flip the inequality sign. Dividing both sides by the coefficient of x is the final step in isolating the variable, allowing us to determine the range of values that satisfy the inequality. This operation is fundamental in solving for x and provides the solution set. Dividing by a positive number preserves the direction of the inequality, ensuring the solution remains accurate.

rac{38}{19} ext{ ≤ } rac{19x}{19}

This simplifies to:

$2 ext{ ≤ } x$

3. Rewrite the Inequality

The inequality $2 ext{ ≤ } x$ can also be written as $x ext{ ≥ } 2$. This is simply a matter of reversing the inequality while maintaining its meaning. This rewriting step is often helpful for clarity and to align with common notations, making the solution easier to interpret and communicate. Both forms of the inequality convey the same information: that x is greater than or equal to 2. This final representation provides a clear and concise understanding of the solution set.

Final Answer

The solution to the inequality $58 - 10x ext{ ≤ } 20 + 9x$ is $x ext{ ≥ } 2$. This means that any value of x that is greater than or equal to 2 will satisfy the original inequality. This solution set includes 2 and all real numbers greater than 2, which can be visualized on a number line as a ray starting at 2 and extending to positive infinity. The ability to solve and interpret inequalities like this is a crucial skill in algebra and beyond, providing the foundation for more advanced mathematical concepts. Understanding the solution set is key to applying these concepts in real-world scenarios and further mathematical studies. This thorough analysis ensures a comprehensive understanding of the solution and its implications.

Therefore, the correct answer is:

• B. $x ext{ ≥ } 2$

Visualizing the Solution on a Number Line

A number line provides a visual representation of the solution set for an inequality. For the solution $x ext{ ≥ } 2$, we would draw a closed circle (or a filled-in dot) at 2 on the number line to indicate that 2 is included in the solution. Then, we would draw an arrow extending to the right from 2, indicating that all numbers greater than 2 are also part of the solution. This visual aid helps to solidify the understanding of the inequality and the range of values that satisfy it. Visualizing the solution set is particularly helpful when dealing with more complex inequalities or systems of inequalities. The number line representation offers a clear and intuitive way to grasp the solution, enhancing comprehension and problem-solving skills. By seeing the solution graphically, you can better understand the range of values that fulfill the inequality.

Common Mistakes to Avoid

When solving inequalities, there are a few common mistakes to watch out for:

• Forgetting to Flip the Inequality Sign: The most common mistake is failing to flip the inequality sign when multiplying or dividing both sides by a negative number. Remember, this is a crucial step to maintain the correctness of the solution. Failing to do so will result in an incorrect solution set. This is a critical rule to remember when manipulating inequalities. Double-checking this step can prevent errors and ensure accuracy.
• Incorrectly Combining Like Terms: Ensure that you are combining like terms correctly. Pay attention to the signs of the terms and make sure you are performing the correct arithmetic operations. A simple arithmetic error can lead to a completely different solution. Taking the time to double-check your work can prevent this type of mistake.
• Misinterpreting the Solution: Make sure you understand what the solution set represents. For example, $x ext{ ≥ } 2$ means all numbers greater than or equal to 2, not just numbers greater than 2. Misinterpreting the solution can lead to incorrect conclusions and applications. Understanding the notation and the meaning of the inequality symbols is crucial for accurate interpretation.

By being aware of these common pitfalls and taking care in each step, you can increase your accuracy and confidence in solving inequalities.

Practice Problems

To solidify your understanding, try solving these similar inequalities:

1. $3x - 7 ext{ > } 5x + 1$
2. $12 - 4x ext{ ≤ } 8 + 2x$
3. $-2(x + 3) ext{ ≥ } 10$

Working through these practice problems will reinforce the steps and concepts discussed in this guide. Practice is essential for mastering any mathematical skill. By applying the techniques you've learned, you'll become more proficient and confident in solving inequalities. These exercises provide an opportunity to test your understanding and identify areas where you may need further clarification. Consistent practice will lead to a deeper understanding and improved problem-solving abilities.

Conclusion

Solving linear inequalities is a fundamental skill in algebra. By following the steps outlined above and being mindful of the rules, you can confidently solve a wide range of inequalities. Remember to combine like terms, isolate the variable, and flip the inequality sign when necessary. With practice, you'll master this important concept and be well-prepared for more advanced mathematical topics. The ability to solve inequalities is not only valuable in mathematics but also in various real-world applications, such as optimization problems, decision-making, and modeling. Mastering this skill provides a solid foundation for future mathematical endeavors and practical problem-solving scenarios.