Solving Compound Inequalities Expressing Solutions In Interval Notation
In mathematics, compound inequalities combine two or more inequalities into a single statement. Solving these inequalities involves isolating the variable while maintaining the truth of the combined statement. The solution is often expressed in interval notation, a concise way to represent a set of numbers within a specific range.
Understanding Compound Inequalities
Compound inequalities typically come in two forms: and inequalities and or inequalities. In the case of and inequalities, both inequalities must be true simultaneously. For or inequalities, at least one of the inequalities must be true. Our focus here is on solving a compound inequality of the and type, specifically: -10 < 3x + 5 ≤ 8.
This inequality states that the expression 3x + 5 must be greater than -10 and less than or equal to 8. To solve this, we aim to isolate x in the middle section of the inequality, performing the same operations on all three parts to maintain balance. The step-by-step solution is detailed below, with explanations to ensure clarity.
Step-by-Step Solution
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Isolate the term with x: To begin, we want to isolate the term containing x, which is 3x. We achieve this by subtracting 5 from all three parts of the inequality:
-10 - 5 < 3x + 5 - 5 ≤ 8 - 5
This simplifies to:
-15 < 3x ≤ 3
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Solve for x: Next, we need to isolate x completely. Since x is multiplied by 3, we divide all three parts of the inequality by 3:
-15 / 3 < 3x / 3 ≤ 3 / 3
This gives us:
-5 < x ≤ 1
This result tells us that x must be greater than -5 and less than or equal to 1. This is the solution to the compound inequality.
Expressing the Solution in Interval Notation
Interval notation is a standardized way to represent intervals of real numbers. It uses parentheses and brackets to indicate whether the endpoints are included or excluded from the interval. The solution we found, -5 < x ≤ 1, can be expressed in interval notation as (-5, 1].
- The parenthesis around -5 indicates that -5 is not included in the solution set; x must be strictly greater than -5. This is because our inequality uses the less than symbol (<).
- The bracket around 1 indicates that 1 is included in the solution set; x can be equal to 1. This corresponds to the less than or equal to symbol (≤) in the inequality.
Therefore, the solution set in interval notation is (-5, 1]. This notation concisely represents all real numbers between -5 and 1, including 1 but excluding -5.
Graphing the Solution on a Number Line
Visualizing the solution on a number line provides another way to understand the interval. To graph the solution (-5, 1], we draw a number line and mark the points -5 and 1. At -5, we use an open circle (or a parenthesis) to indicate that it is not included in the solution. At 1, we use a closed circle (or a bracket) to show that it is included. Then, we shade the region between -5 and 1 to represent all the numbers in the solution set.
The graph clearly illustrates that the solution includes all numbers from just above -5 up to and including 1. This visual representation can be particularly helpful for students who are learning about inequalities and interval notation, as it connects the algebraic solution to a visual concept.
Common Mistakes and How to Avoid Them
When solving compound inequalities, it’s crucial to avoid common pitfalls to ensure accurate results. One frequent error is failing to perform the same operation on all parts of the inequality. Remember, whatever you do to one part, you must do to all parts to maintain the balance and validity of the inequality.
Another mistake is misinterpreting the symbols and their implications for interval notation. A less than (<) or greater than (>) symbol indicates that the endpoint is not included, requiring a parenthesis in interval notation. A less than or equal to (≤) or greater than or equal to (≥) symbol means the endpoint is included, necessitating a bracket. Paying close attention to these symbols is essential for correct interval notation.
Finally, be mindful of the order when writing interval notation. The smaller number always comes first, followed by the larger number. Reversing the order leads to an incorrect representation of the interval.
By understanding these common errors and actively working to avoid them, students can improve their accuracy and confidence in solving compound inequalities.
Real-World Applications of Compound Inequalities
Compound inequalities are not just abstract mathematical concepts; they have practical applications in various real-world scenarios. One common application is in determining acceptable ranges for values. For example, in manufacturing, product dimensions must often fall within a specific range to meet quality standards. This can be expressed using a compound inequality.
Consider a scenario where a machine produces metal rods, and the acceptable length is between 10.5 cm and 11.0 cm, inclusive. This can be represented as the compound inequality 10.5 ≤ length ≤ 11.0. Any rod outside this range would be considered defective.
In finance, compound inequalities can be used to define investment criteria. For instance, an investor might specify that they only want to invest in companies with a price-to-earnings ratio between 10 and 15. This can be expressed as 10 ≤ P/E ratio ≤ 15.
In everyday life, compound inequalities can describe temperature ranges for optimal comfort, speed limits on roads, or even the number of calories one should consume daily for a healthy diet. These examples illustrate that compound inequalities are versatile tools for setting boundaries and defining acceptable conditions in various contexts.
By recognizing these real-world applications, students can better appreciate the relevance and importance of learning how to solve compound inequalities. This understanding can also make the mathematical concepts more engaging and easier to grasp.
Conclusion
Solving the compound inequality -10 < 3x + 5 ≤ 8 involves isolating the variable x and expressing the solution in interval notation. By following the steps of subtracting 5 and then dividing by 3, we found the solution -5 < x ≤ 1. This is represented in interval notation as (-5, 1], indicating all numbers greater than -5 and less than or equal to 1. This process demonstrates the importance of careful algebraic manipulation and the power of interval notation in expressing solution sets. Understanding compound inequalities is a fundamental skill in algebra with numerous applications in various fields.
Solve the compound inequality -10 < 3x + 5 ≤ 8 and express the solution in interval notation.
Solving Compound Inequalities Expressing Solutions in Interval Notation
