What are 'And' Compound Inequalities? Definition and Examples

📚 Quick Study Guide

🔍 'And' compound inequalities combine two inequalities with an 'and' statement, meaning both must be true. 💡 They can be written as $a < x < b$, which means $a < x$ and $x < b$. 📝 To solve, isolate the variable in the middle of the compound inequality. ➕ Perform the same operation on all three parts (left, middle, and right) to maintain the inequality. 📈 The solution is the intersection of the solutions to the individual inequalities. 📊 Graphically, the solution is the region where the graphs of the two inequalities overlap. 🧭 Examples: Solve $ -3 < 2x + 1 < 7$ or $x > 2$ and $x < 5$.

Practice Quiz

1. Which of the following represents an 'and' compound inequality?
1. A) $x < 3$ or $x > 5$
2. B) $x < 3$ and $x > 5$
3. C) $x = 3$
4. D) $x > 5$
2. What does the solution to an 'and' compound inequality represent?
1. A) All values that satisfy at least one of the inequalities.
2. B) All values that satisfy both inequalities simultaneously.
3. C) All values that satisfy neither inequality.
4. D) The average of the solutions to each inequality.
3. Solve the compound inequality: $2 < x + 3 < 5$
1. A) $-1 < x < 2$
2. B) $5 < x < 8$
3. C) $-1 > x > 2$
4. D) $x < -1$ or $x > 2$
4. Which graph represents the solution to $1 \le x \le 4$?
1. A) A number line with open circles at 1 and 4, and the region between them shaded.
2. B) A number line with closed circles at 1 and 4, and the region between them shaded.
3. C) A number line with closed circles at 1 and 4, and the regions to the left of 1 and to the right of 4 shaded.
4. D) A number line with open circles at 1 and 4, and the regions to the left of 1 and to the right of 4 shaded.
5. What is the first step in solving the compound inequality $-5 < 3x - 2 < 1$?
1. A) Divide all parts by 3.
2. B) Subtract 2 from all parts.
3. C) Add 2 to all parts.
4. D) Multiply all parts by -5.
6. Which of the following is equivalent to the compound inequality $x > -2$ and $x < 3$?
1. A) $x > 3$
2. B) $x < -2$
3. C) $-2 < x < 3$
4. D) $x < -2$ or $x > 3$
7. What is the solution to $x + 1 < 2$ and $x - 1 > 0$?
1. A) $0 < x < 1$
2. B) $1 < x < 1$
3. C) $1 < x < 2$
4. D) No solution