Solving 2(5-x) ≤ 14 A Step-by-Step Guide To Inequalities
Understanding Inequalities
Before diving into the specifics of our problem, it's essential to grasp the basic concept of inequalities. Unlike equations, which show that two expressions are equal, inequalities express a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another. These relationships are represented by the symbols >, <, ≥, and ≤, respectively. Solving an inequality means finding the range of values for a variable that makes the inequality true. This often involves algebraic manipulations similar to those used in solving equations, but with a few key differences that we'll highlight.
When working with inequalities, it's crucial to remember that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. This is a critical rule that distinguishes inequality solving from equation solving. For instance, if we have -x < 5, multiplying both sides by -1 gives us x > -5. Neglecting this rule can lead to incorrect solutions. Additionally, understanding the properties of real numbers and how they interact with inequalities is fundamental. For example, adding or subtracting the same number from both sides of an inequality preserves the inequality, a principle we will use in our solution.
Furthermore, visualizing inequalities on a number line can provide a deeper understanding of the solution set. A solution like x ≥ 2 represents all numbers greater than or equal to 2, which can be illustrated on a number line with a closed circle at 2 and a line extending to the right. Similarly, x < 3 represents all numbers less than 3, shown with an open circle at 3 and a line extending to the left. This visual representation can be particularly helpful when dealing with compound inequalities or intervals.
Step-by-Step Solution of 2(5-x) ≤ 14
Let's tackle the inequality 2(5-x) ≤ 14 step-by-step. Our goal is to isolate the variable x on one side of the inequality. Here’s how we proceed:
1. Distribute the 2
The first step involves distributing the 2 on the left side of the inequality. This means multiplying both terms inside the parentheses by 2:
2 * 5 - 2 * x ≤ 14
This simplifies to:
10 - 2x ≤ 14
Distribution is a fundamental algebraic operation that allows us to remove parentheses and simplify expressions. In this case, it transforms the original inequality into a more manageable form. Accurate distribution is crucial; any error here will propagate through the rest of the solution. It is important to double-check this step to ensure that the multiplication is performed correctly for each term within the parentheses.
2. Isolate the Term with x
Next, we want to isolate the term containing x. To do this, we subtract 10 from both sides of the inequality. Remember, subtracting the same number from both sides preserves the inequality:
10 - 2x - 10 ≤ 14 - 10
This simplifies to:
-2x ≤ 4
Isolating the variable term is a common strategy in solving both equations and inequalities. By subtracting 10, we effectively move the constant term to the right side of the inequality, bringing us closer to isolating x. This step relies on the property that adding or subtracting the same value from both sides of an inequality does not change the direction of the inequality sign.
3. Solve for x
Now, we need to solve for x by dividing both sides of the inequality by -2. This is a crucial step where we must remember the rule about reversing the inequality sign when dividing by a negative number:
(-2x) / -2 ≥ 4 / -2
Notice that the “≤” sign has changed to “≥”. This is because we divided by a negative number. The inequality now simplifies to:
x ≥ -2
Dividing by a negative number and flipping the inequality sign is perhaps the most critical rule to remember when solving inequalities. This is where many mistakes can occur if this rule is overlooked. Understanding why this rule exists is important: dividing by a negative number essentially reflects the number line, so the order of the numbers is reversed. Therefore, we must reverse the inequality sign to maintain the truth of the statement.
4. Interpret the Solution
The solution x ≥ -2 means that any value of x that is greater than or equal to -2 will satisfy the original inequality. This includes -2, -1, 0, 1, 2, and so on. We can represent this solution on a number line by shading the region to the right of -2, including -2 itself (indicated by a closed circle at -2).
Interpreting the solution is as important as finding it. Understanding what the solution means in the context of the original problem allows us to verify if the solution is reasonable and makes sense. In this case, x ≥ -2 represents an infinite set of values that all satisfy the given inequality. This understanding is also crucial when dealing with more complex problems, such as systems of inequalities or applications of inequalities in real-world scenarios.
Choosing the Correct Answer
Based on our step-by-step solution, we have determined that x ≥ -2. Now, let’s look at the options provided:
A) x ≤ 2 B) x ≥ 2 C) x ≥ -2 D) x ≤ 1
The correct answer is C) x ≥ -2. This matches the solution we derived through our algebraic manipulations.
Selecting the correct answer involves comparing our solution to the given options and identifying the one that matches exactly. It is important to carefully review each option and ensure that it aligns with the solution we have obtained. This step serves as a final check to confirm the accuracy of our work.
Common Mistakes to Avoid
When solving inequalities, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them.
- Forgetting to Reverse the Inequality Sign: As highlighted earlier, this is the most common mistake. Always remember to reverse the inequality sign when multiplying or dividing both sides by a negative number.
- Incorrect Distribution: Errors in distribution can significantly alter the inequality. Double-check that you have multiplied each term inside the parentheses correctly.
- Arithmetic Errors: Simple arithmetic mistakes, such as adding or subtracting incorrectly, can lead to wrong answers. Take your time and verify each calculation.
- Misinterpreting the Solution: Understanding what the solution represents is crucial. For example, x > 3 is different from x ≥ 3. The former excludes 3, while the latter includes it.
- Not Checking the Solution: Plugging a value from your solution set back into the original inequality can help you verify if your solution is correct. This is a valuable step in ensuring accuracy.
By being mindful of these common mistakes, you can significantly improve your accuracy in solving inequalities.
Practice Problems
To solidify your understanding, here are a few practice problems:
- Solve: 3(2 - x) < 9
- Solve: -4(x + 1) ≥ 8
- Solve: 5x - 7 ≤ 3x + 1
Working through these problems will reinforce the steps and concepts discussed in this article. Remember to apply the rules and techniques we've covered, and don't hesitate to review the steps if you encounter difficulties.
Conclusion
Solving inequalities is a vital skill in mathematics with numerous applications. In this article, we provided a comprehensive guide on how to solve the inequality 2(5-x) ≤ 14. By following a step-by-step approach, remembering the key rule about reversing the inequality sign, and avoiding common mistakes, you can confidently solve a wide range of inequalities. Practice is essential to mastering this skill, so be sure to work through additional problems to enhance your understanding and proficiency. Mastering inequalities opens doors to more advanced mathematical concepts and real-world problem-solving scenarios. The solution to the inequality 2(5-x) ≤ 14 is x ≥ -2, and by understanding the process, you're well-equipped to tackle similar challenges.
