Solving The Inequality -5 < -7 - 2x < 4 A Step-by-Step Guide
This article provides a step-by-step guide on how to solve the compound inequality . Understanding how to manipulate inequalities is crucial in various fields of mathematics, including algebra, calculus, and real analysis. This particular inequality involves a variable, , and we aim to isolate to find the range of values that satisfy the given condition. By the end of this guide, you will have a clear understanding of the process involved in solving this type of inequality and be able to apply similar techniques to other problems.
Understanding Compound Inequalities
Compound inequalities are mathematical statements that combine two or more inequalities using conjunctions such as "and" or disjunctions such as "or." In the given problem, we have a compound inequality that can be read as "-5 is less than -7 - 2x, and -7 - 2x is less than 4." To solve this type of inequality, we need to isolate the variable by performing algebraic operations on all parts of the inequality. The key principle to remember is that any operation performed on one part of the inequality must be performed on all parts to maintain the inequality's balance. This includes adding, subtracting, multiplying, or dividing by constants. However, when multiplying or dividing by a negative number, the direction of the inequality signs must be reversed. Understanding this fundamental rule is crucial for correctly solving inequalities and avoiding common mistakes. By grasping the concept of compound inequalities and the rules governing their manipulation, you can confidently tackle a wide range of mathematical problems involving inequalities.
Step-by-Step Solution
To solve the inequality , we will isolate by performing the same operations on all three parts of the inequality. This ensures that the balance of the inequality is maintained throughout the solution process. Let's break down the steps:
Step 1: Add 7 to all parts of the inequality
We begin by adding 7 to all parts of the inequality to eliminate the -7 term in the middle. This operation helps to simplify the expression and bring us closer to isolating the term containing . Adding 7 to all parts gives us:
This simplifies to:
Step 2: Divide all parts of the inequality by -2
Next, we need to divide all parts of the inequality by -2 to isolate . It's crucial to remember that when we divide by a negative number, we must reverse the direction of the inequality signs. This is a fundamental rule in inequality manipulation and is essential for obtaining the correct solution. Dividing by -2 gives us:
rac{2}{-2} > rac{-2x}{-2} > rac{11}{-2}
This simplifies to:
-1 > x > -rac{11}{2}
Step 3: Rewrite the inequality
To express the solution in a more conventional format, we can rewrite the inequality so that the smaller value is on the left and the larger value is on the right. This makes the solution easier to interpret and understand. Rewriting the inequality gives us:
-rac{11}{2} < x < -1
This inequality states that is greater than -rac{11}{2} and less than -1. In decimal form, this is -5.5 < x < -1.
Solution
The solution to the inequality is -rac{11}{2} < x < -1. This means that any value of between -5.5 and -1 (exclusive) will satisfy the original inequality. We can represent this solution on a number line by drawing an open interval between -5.5 and -1, indicating that the endpoints are not included in the solution set. The solution can also be expressed in interval notation as . Understanding how to express the solution in different forms is important for clear communication and interpretation of mathematical results. This solution provides a range of values for that make the given inequality true, demonstrating the power of algebraic manipulation in solving mathematical problems. By following the steps outlined above, you can confidently solve similar compound inequalities and gain a deeper understanding of mathematical concepts.
Verification
To verify the solution, we can pick a value within the interval -rac{11}{2} < x < -1 and substitute it into the original inequality. If the inequality holds true, it confirms that our solution is correct. Let's choose , which falls within the interval (-5.5, -1).
Substituting into the original inequality , we get:
Simplifying the middle part, we have:
This statement is true, as -1 is indeed greater than -5 and less than 4. This confirms that is a valid solution, and our interval -rac{11}{2} < x < -1 is likely correct. To further ensure the accuracy of our solution, we can also test values outside the interval. If we choose a value less than -5.5, such as , and substitute it into the original inequality, we get:
This statement is false, as 5 is not less than 4. Similarly, if we choose a value greater than -1, such as , we get:
This statement is also false, as -7 is not greater than -5. These tests confirm that our solution interval -rac{11}{2} < x < -1 is indeed the correct range of values for that satisfy the original inequality. By verifying the solution with multiple test values, we can have confidence in the accuracy of our answer.
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 and ensure accurate results. One of the most frequent errors is forgetting to reverse the direction of the inequality signs when multiplying or dividing by a negative number. As demonstrated in the step-by-step solution, this reversal is crucial for maintaining the correct relationship between the expressions. Another common mistake is not applying the same operation to all parts of the inequality. To keep the inequality balanced, any operation performed on one part must be performed on all parts. This includes addition, subtraction, multiplication, and division. Failing to do so can alter the inequality and lead to an incorrect solution set. Additionally, errors can occur when simplifying expressions or combining like terms. It's essential to double-check each step to ensure that algebraic manipulations are performed correctly. Careless mistakes in arithmetic can quickly lead to an incorrect answer. Finally, misinterpreting the solution set is another potential pitfall. It's important to understand whether the endpoints of the interval are included or excluded, and to express the solution in the correct notation (interval notation or inequality notation). By being mindful of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving inequalities.
Conclusion
In conclusion, solving the inequality involves isolating the variable by performing algebraic operations on all parts of the inequality. We began by adding 7 to all parts, followed by dividing by -2 (and reversing the inequality signs), and finally, rewriting the inequality in a standard format. The solution obtained is -rac{11}{2} < x < -1, which represents the range of values for that satisfy the given condition. This solution was verified by substituting a value within the interval into the original inequality and confirming that the statement holds true. Furthermore, we discussed common mistakes to avoid, such as forgetting to reverse inequality signs when multiplying or dividing by a negative number and not applying operations to all parts of the inequality. By understanding the steps involved and being mindful of potential errors, you can confidently solve similar inequalities and apply these techniques to more complex mathematical problems. The ability to manipulate and solve inequalities is a valuable skill in various areas of mathematics and its applications.