📚 Definition of Compound Inequalities with "And"
A compound inequality with "and" combines two inequalities into a single statement, where both inequalities must be true simultaneously. This creates an intersection of the solution sets for each individual inequality.
📜 History and Background
The concept of inequalities has ancient roots, appearing in early mathematical texts. Compound inequalities, building upon this foundation, emerged as mathematicians sought to express more complex relationships and constraints. They became particularly important in fields like optimization and linear programming.
🔑 Key Principles
- 🔍Intersection: The solution to a compound inequality with "and" is the intersection of the solutions to each individual inequality. This means the value of $x$ must satisfy *both* inequalities.
- 📈Graphical Representation: On a number line, the solution is represented by the region where the solutions to both inequalities overlap.
- 🧮Algebraic Manipulation: To solve, isolate the variable in each inequality separately and then find the intersection of the solutions.
- 📝Symbolic Representation: A compound inequality with "and" can be written as $a < x$ AND $x < b$, which is often abbreviated as $a < x < b$.
➕ Solving Compound Inequalities with "And" - Example
Let's solve the compound inequality: $2 < x + 1$ AND $x + 1 < 5$
- Isolate x in each inequality:
$2 < x + 1$ becomes $1 < x$
$x + 1 < 5$ becomes $x < 4$
- Combine the inequalities: $1 < x < 4$
- Solution: The solution set is all $x$ such that $x$ is greater than 1 and less than 4.
🌍 Real-world Examples
- 🌡️ Temperature Range: A chemical reaction may require a temperature that is greater than 20°C AND less than 30°C. This can be expressed as $20 < T < 30$, where $T$ is the temperature.
- 📏 Manufacturing Tolerances: A machine part must have a length greater than 2.5 cm AND less than 2.6 cm to be considered acceptable. This is written as $2.5 < L < 2.6$, where $L$ is the length.
- 🌱 Plant Growth: A plant needs a pH level greater than 6.0 AND less than 7.0 to thrive. Expressed as $6.0 < pH < 7.0$.
💡 Conclusion
Understanding compound inequalities with "and" is crucial for problem-solving in algebra and beyond. By remembering that the solution must satisfy *both* conditions, you can confidently tackle these types of problems.