Solving Compound Inequalities $-13 \\leq 2y - 3 < -9$ With Interval Notation

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In mathematics, compound inequalities combine two or more inequalities using the words "and" or "or." Solving compound inequalities involves isolating the variable while maintaining the integrity of the inequality. The solution is often expressed in interval notation, a concise way to represent a set of real numbers.

Understanding Compound Inequalities

A compound inequality is essentially two inequalities joined by either "and" or "or." When dealing with an "and" compound inequality, we seek values that satisfy both inequalities simultaneously. Conversely, an "or" compound inequality seeks values that satisfy at least one of the inequalities.

For instance, the inequality −13leq2y−3<−9-13 \\leq 2y - 3 < -9 is a compound inequality using "and." It can be interpreted as two separate inequalities:

  1. −13leq2y−3-13 \\leq 2y - 3
  2. 2y−3<−92y - 3 < -9

The solution to this compound inequality will be the set of values that satisfy both inequalities simultaneously.

Solving the Compound Inequality −13≤2y−3<−9-13 \\\\\leq 2y - 3 < -9

To solve the compound inequality −13leq2y−3<−9-13 \\leq 2y - 3 < -9, our goal is to isolate the variable y. We can achieve this by performing the same operations on all parts of the inequality. The key is to maintain the balance of the inequality throughout the process.

Step 1: Isolate the Term with y

Our first step is to isolate the term containing y, which in this case is 2y2y. To do this, we need to eliminate the constant term, -3, from the middle part of the inequality. We can accomplish this by adding 3 to all three parts of the inequality:

−13+3leq2y−3+3<−9+3-13 + 3 \\leq 2y - 3 + 3 < -9 + 3

This simplifies to:

−10leq2y<−6-10 \\leq 2y < -6

Step 2: Isolate y

Now that we have isolated the term with y, we need to isolate y itself. The term 2y2y means 2 multiplied by y. To undo this multiplication, we divide all parts of the inequality by 2:

−102leq2y2<−62\frac{-10}{2} \\leq \frac{2y}{2} < \frac{-6}{2}

This simplifies to:

−5leqy<−3-5 \\leq y < -3

Step 3: Express the Solution in Interval Notation

We have now solved the compound inequality, and the solution is −5leqy<−3-5 \\leq y < -3. This means that y can be any value greater than or equal to -5 and less than -3. To express this solution in interval notation, we use brackets and parentheses. A bracket indicates that the endpoint is included in the solution, while a parenthesis indicates that the endpoint is not included.

In this case, the solution includes -5 (because of the "less than or equal to" sign) and does not include -3 (because of the "less than" sign). Therefore, the interval notation for the solution is:

[−5,−3)[-5, -3)

This interval notation represents all real numbers between -5 and -3, including -5 but not -3.

Interval Notation: A Powerful Tool

Interval notation is a standardized way to represent sets of real numbers. It uses brackets and parentheses to indicate whether the endpoints of an interval are included or excluded. Understanding interval notation is crucial for expressing solutions to inequalities and working with various mathematical concepts.

Here's a quick recap of the symbols used in interval notation:

  • [ ] (Square brackets): Indicate that the endpoint is included in the interval.
  • () (Parentheses): Indicate that the endpoint is not included in the interval.
  • ∞ (Infinity): Represents positive infinity. It is always enclosed in a parenthesis because infinity is not a specific number and cannot be included in an interval.
  • -∞ (Negative infinity): Represents negative infinity. It is also always enclosed in a parenthesis.

For example:

  • (2,5)(2, 5) represents all real numbers between 2 and 5, not including 2 and 5.
  • [2,5][2, 5] represents all real numbers between 2 and 5, including 2 and 5.
  • [2,5)[2, 5) represents all real numbers between 2 and 5, including 2 but not 5.
  • (2,∞)(2, ∞) represents all real numbers greater than 2.
  • (−∞,5](-∞, 5] represents all real numbers less than or equal to 5.

Common Mistakes to Avoid When Solving Compound Inequalities

When solving compound inequalities, it's essential to be mindful of potential errors. Here are some common mistakes to avoid:

  1. Forgetting to Apply Operations to All Parts: When performing an operation (addition, subtraction, multiplication, or division) on a compound inequality, it's crucial to apply it to all parts of the inequality. Failing to do so will disrupt the balance of the inequality and lead to an incorrect solution.

  2. Flipping the Inequality Sign When Multiplying or Dividing by a Negative Number: This is a critical rule to remember. When multiplying or dividing all parts of an inequality by a negative number, you must flip the direction of the inequality signs. For example, if you have -2y < 6, dividing both sides by -2 gives y > -3 (the inequality sign flipped).

  3. Incorrectly Interpreting "And" and "Or": Compound inequalities involving "and" require that both inequalities be satisfied simultaneously. Compound inequalities involving "or" require that at least one of the inequalities be satisfied. Confusing these can lead to incorrect solutions.

  4. Misinterpreting Interval Notation: Understanding the meaning of brackets and parentheses in interval notation is crucial. Misinterpreting them can lead to representing the solution set incorrectly.

Real-World Applications of Compound Inequalities

Compound inequalities aren't just abstract mathematical concepts; they have practical applications in various real-world scenarios. Understanding how to solve and interpret them can be valuable in fields like:

  1. Finance: Compound inequalities can be used to define ranges for investment returns, interest rates, or loan amounts. For instance, a financial advisor might describe a moderate-risk investment as one with an expected annual return between 5% and 10%.

  2. Science: In scientific experiments, compound inequalities can be used to express acceptable ranges for temperature, pressure, or other variables. For example, a chemical reaction might require a temperature between 20°C and 30°C for optimal results.

  3. Engineering: Engineers use compound inequalities to specify tolerances for dimensions, stress levels, or other parameters in designs. This ensures that components function correctly and safely.

  4. Statistics: Compound inequalities can be used to define confidence intervals or other statistical ranges. For example, a poll might report a margin of error, indicating that the true population value is likely to fall within a certain range.

  5. Everyday Life: Compound inequalities can even be used in everyday situations, such as setting goals for exercise or managing budgets. For instance, someone might set a goal to exercise for at least 30 minutes but no more than 60 minutes per day.

Conclusion

Solving compound inequalities is a fundamental skill in algebra and has wide-ranging applications. By understanding the steps involved in isolating the variable and expressing the solution in interval notation, you can effectively tackle these types of problems. Remember to pay attention to the details, such as flipping the inequality sign when multiplying or dividing by a negative number, and correctly interpreting the "and" and "or" conditions. With practice, you'll become proficient in solving compound inequalities and applying them to real-world situations. The solution set to the inequality −13leq2y−3<−9-13 \\leq 2y - 3 < -9 is [−5,−3)[-5, -3).