Definition of 'and' compound inequalities for high school math students

📚 Definition of 'And' Compound Inequalities

An 'and' compound inequality is a statement that combines two inequalities with the word "and." This means that both inequalities must be true at the same time. The solution set consists of all values that satisfy both inequalities simultaneously. Often, these can be written as a single inequality where a variable is between two values.

🕰️ History and Background

The concept of inequalities has been around for centuries, evolving alongside the development of algebra. The formalization of compound inequalities, particularly with the conjunction 'and,' became more prominent as mathematical notation and logic became standardized. This allowed for clearer and more concise expression of conditions requiring multiple constraints.

✨ Key Principles

• ➕ Intersection: The solution to an 'and' compound inequality is the intersection of the solutions to each individual inequality. This means you're looking for the values that are common to both solution sets.
• ✍️ Solving: To solve an 'and' compound inequality, you solve each inequality separately.
• 📊 Graphing: The graph of an 'and' compound inequality is the overlap of the graphs of each individual inequality. This will often be a line segment between two points on a number line.
• 🤝 Equivalent Forms: An 'and' compound inequality like $x > a \text{ and } x < b$ (where $a < b$) can be written in the more compact form $a < x < b$. This form emphasizes that $x$ must lie between $a$ and $b$.

📝 Real-world Examples

Let's look at some examples to make this clearer:

Example 1:

Solve and graph the compound inequality: $x > -2 \text{ and } x \le 3$

Solution:

The solution includes all numbers greater than -2 and less than or equal to 3. On a number line, this is the interval between -2 (not included) and 3 (included). In interval notation, this is $(-2, 3]$.

Example 2:

Solve and graph the compound inequality: $-1 \le 2x + 1 < 5$

Solution:

This is an 'and' compound inequality written in compact form. To solve, we isolate $x$ by performing the same operations on all parts of the inequality:

Subtract 1 from all parts: $-1 - 1 \le 2x + 1 - 1 < 5 - 1$ which simplifies to $-2 \le 2x < 4$

Divide all parts by 2: $\frac{-2}{2} \le \frac{2x}{2} < \frac{4}{2}$ which simplifies to $-1 \le x < 2$

The solution is all numbers greater than or equal to -1 and less than 2. In interval notation, this is $[-1, 2)$.

✍️ Practice Quiz

Test your knowledge! Solve the following 'and' compound inequalities:

1. ❓ $x + 3 > 1 \text{ and } x - 2 < 0$
2. ❓ $-3 \le x - 1 \le 2$
3. ❓ $2x + 1 > -5 \text{ and } 3x - 2 \le 7$

Answers:

1. ✅ $x > -2 \text{ and } x < 2$, or $(-2, 2)$
2. ✅ $-2 \le x \le 3$, or $[-2, 3]$
3. ✅ $x > -3 \text{ and } x \le 3$, or $(-3, 3]$

💡 Conclusion

'And' compound inequalities require both inequalities to be true simultaneously. By understanding the concept of intersection and practicing with various examples, you can confidently solve and graph these inequalities. Keep practicing, and you'll master them in no time! 🎉