Verified Answer :
- 17: \(b > 3\)
- 18: \(r > 7\) or \(r \leq -7\)
- 19: \(x < 6\)
- 20: \(s \in \mathbb{R}\)
- 21: \(x < 2.5\)
- 22: \(2 < y < 5\)
- 23: \(\emptyset\)
- 24: \(x \in \mathbb{R}\)
- 25: \(n \in \mathbb{R}\)
- 26: \(0.5 < x \leq 2\)
Final answers are: \(b > 3\); \(r > 7 \text{ or } r \leq -7\); \(x < 6\); \(s \in \mathbb{R}\); \(x < 2.5\); \(2 < y < 5\); \(\emptyset\); \(x \in \mathbb{R}\); \(n \in \mathbb{R}\); \(0.5 < x \leq 2\) \(\boxed{b > 3; r > 7 \text{ or } r \leq -7; x < 6; \mathbb{R}; x < 2.5; 2 < y < 5; \emptyset; \mathbb{R}; \mathbb{R}; 0.5 < x \leq 2}\)
Explanation
- Analysis and Strategy
To solve compound inequalities, we use the properties of logical connectors: "or" corresponds to the union of solution sets, and "and" corresponds to the intersection of solution sets. For a chain inequality like \(A > B > C\), we treat it as the conjunction of two separate inequalities \(A > B\) and \(B > C\). For each sub-problem, we will isolate the variable on one side and combine the solutions according to these principles.
- Solving 17-20
- \(2b - 6 \geq 14\) or \(8b + 2b - 3 > 27\)
\(2b \geq 20 \implies b \geq 10\).
\(10b > 30 \implies b > 3\).
The union of \(b \geq 10\) and \(b > 3\) is \(b > 3\).
- \(r - 7 > 0\) or \(2r + 16 \leq 2\)
\(r > 7\).
\(2r \leq -14 \implies r \leq -7\).
The union is \(r > 7\) or \(r \leq -7\).
- \(3x + 6 \leq -9\) or \(x < 6\)
\(3x \leq -15 \implies x \leq -5\).
\(x < 6\).
The union of \(x \leq -5\) and \(x < 6\) is \(x < 6\).
- \(2s - 7 > -21\) or \(4s + 6 eq 22\)
\(2s > -14 \implies s > -7\).
\(4s
eq 16 \implies s
eq 4\).
The union of \(s > -7\) and \(s
eq 4\) is all real numbers, since any real number is either greater than -7 or not equal to 4. Thus, \(s \in \mathbb{R}\).
- Solving 21-23
- \(3 - 2x > -2\) or \(3x + 6 < 8\)
\(-2x > -5 \implies x < 2.5\).
\(3x < 2 \implies x < \frac{2}{3}\).
The union of \(x < 2.5\) and \(x < \frac{2}{3}\) is \(x < 2.5\).
- \(7y > 2y + 10\) and \(2y > 5y - 15\)
\(5y > 10 \implies y > 2\).
\(-3y > -15 \implies y < 5\).
The intersection of \(y > 2\) and \(y < 5\) is \(2 < y < 5\).
- \(4m - 4 \leq 6m - 8\) and \(-3m + 1 \geq 5m - 3\)
\(-2m \leq -4 \implies m \geq 2\).
\(-8m \geq -4 \implies m \leq 0.5\).
The intersection of \(m \geq 2\) and \(m \leq 0.5\) is the empty set, \(\emptyset\).
- Solving 24-26
- \(4x + 6 > 10 - x\) or \(9 - 3x \geq -1\)
\(5x > 4 \implies x > 0.8\).
\(-3x \geq -10 \implies x \leq \frac{10}{3}\).
The union of \(x > 0.8\) and \(x \leq \frac{10}{3}\) covers all real numbers because every \(x\) is either \(x > 0.8\) or \(x \leq 0.8\) (and \(x \leq 0.8\) is a subset of \(x \leq \frac{10}{3}\)). Thus, \(x \in \mathbb{R}\).
- \(3n - 2 < 4\) or \(2n + 8 < 5n + 11\)
\(3n < 6 \implies n < 2\).
\(-3n < 3 \implies n > -1\).
The union of \(n < 2\) and \(n > -1\) is all real numbers, \(n \in \mathbb{R}\).
- \(6 - 3x \geq 4 - 2x > 5 - 4x\)
This is the conjunction of \(6 - 3x \geq 4 - 2x\) and \(4 - 2x > 5 - 4x\).
\(6 - 3x \geq 4 - 2x \implies -x \geq -2 \implies x \leq 2\).
\(4 - 2x > 5 - 4x \implies 2x > 1 \implies x > 0.5\).
The intersection is \(0.5 < x \leq 2\).
Examples
Compound inequalities appear in physics when determining the safe operating range of a machine. For example, a robotic arm may need to maintain a temperature \(T\) such that \(T > 20^\circ C\) and \(T < 50^\circ C\) to function without overheating or freezing. By solving these as a single compound inequality \(20 < T < 50\), engineers can set automated control systems to keep the equipment within its optimal working bounds.