- Algebraic inequalities: 25: \(n \in \mathbb{R}\); 26: \(0.5 < x \leq 2\); 27: \(z \leq -2.5\) or \(z > 3\); 28: \(c > 6\).
- Word problems/Physics: 29: \(t < 125\) or \(t > 160\); 30: \(s < 45\) or \(s > 65\); 31: \(10^{16} \leq f \leq 10^{19}\); 32: \(f < 4.3 \times 10^{14}\) or \(f > 7.5 \times 10^{14}\); 33: \(\lambda < 10\) or \(\lambda > 400\); 34: \(1 \text{ mm} \leq \lambda \leq 1 \text{ m}\).
- Complements: 35: \(x < -3\) or \(x > 1\); 36: \(2 \leq x \leq 4\); 37: \(-5 < x \leq 0\); 38: \(x \leq -8\) or \(x > 13\).
- Conceptual: 39: Conjunction represents the intersection (overlap) of two conditions; disjunction represents the union (totality) of two conditions.
Explanation
- Solving Inequalities and Formulating Solutions
We have several tasks involving inequalities. Let's solve them step by step.
- Solve \(3n - 2 < 4\) or \(2n + 8 < 5n + 11\).
First part: \(3n - 2 < 4 \implies 3n < 6 \implies n < 2\).
Second part: \(2n + 8 < 5n + 11 \implies 8 - 11 < 5n - 2n \implies -3 < 3n \implies -1 < n\), which is \(n > -1\).
Since this is a disjunction ("or"), we look at the union of \(n < 2\) and \(n > -1\).
For any real number \(n\), it satisfies either \(n < 2\) or \(n > -1\). Thus, the solution is all real numbers, denoted as \((-\infty, \infty)\).
- Solve \(6 - 3x \geq 4 - 2x > 5 - 4x\).
This is a conjunction ("and"). We solve the two inequalities:
\(6 - 3x \geq 4 - 2x \implies 6 - 4 \geq 3x - 2x \implies 2 \geq x\), or \(x \leq 2\).
\(4 - 2x > 5 - 4x \implies 4x - 2x > 5 - 4 \implies 2x > 1 \implies x > 0.5\).
The intersection is \(0.5 < x \leq 2\).
- Solve \(2z + 3 \leq -2\) or \(3z - 5 > 4\).
First part: \(2z + 3 \leq -2 \implies 2z \leq -5 \implies z \leq -2.5\).
Second part: \(3z - 5 > 4 \implies 3z > 9 \implies z > 3\).
The union is \(z \leq -2.5\) or \(z > 3\).
- Solve \(2c + 50 < 7c + 2 + 3c\) and \(5 - 4c < 53\).
First part: \(2c + 50 < 10c + 2 \implies 48 < 8c \implies c > 6\).
Second part: \(5 - 4c < 53 \implies -4c < 48 \implies c > -12\).
The intersection of \(c > 6\) and \(c > -12\) is \(c > 6\).
- Undesirable temperatures \(t\) for beef: \(t < 125^{\circ}\text{F}\) or \(t > 160^{\circ}\text{F}\).
- Illegal speeds \(s\) on I-85: \(s < 45\text{ mi/hr}\) or \(s > 65\text{ mi/hr}\).
- Frequency of x-rays (\(10^{16} \text{ Hz}\) to \(10^{19} \text{ Hz}\)): \(10^{16} \text{ Hz} \leq f \leq 10^{19} \text{ Hz}\).
- Frequency of nonvisible radiation (not in visible range \(4.3 \times 10^{14} \text{ Hz}\) to \(7.5 \times 10^{14} \text{ Hz}\)): \(f < 4.3 \times 10^{14} \text{ Hz}\) or \(f > 7.5 \times 10^{14} \text{ Hz}\).
- Wavelength of non-ultraviolet radiation (ultraviolet is \(10 \text{ nm}\) to \(400 \text{ nm}\), non-ultraviolet is outside this): \(\lambda < 10 \text{ nm}\) or \(\lambda > 400 \text{ nm}\).
- Wavelength of microwaves (\(1 \text{ mm}\) to \(1 \text{ m}\)): \(1 \text{ mm} \leq \lambda \leq 1 \text{ m}\).
- Complement of \(-3 \leq x \leq 1\): \(x < -3\) or \(x > 1\).
- Complement of \(x > 4\) or \(x < 2\): \(2 \leq x \leq 4\).
- Complement of \(x \leq -5\) or \(x > 0\): \(-5 < x \leq 0\).
- Complement of \(x > -8\) and \(x \leq 13\): \(x \leq -8\) or \(x > 13\).
- Conjunction ("and") finds the overlap between two sets of numbers on a graph, while a disjunction ("or") identifies the entire combined set of numbers represented by the two graphs.
Examples
Imagine you are managing an assembly line where robots must operate within specific speed and temperature ranges. If the temperature exceeds safe limits, the system triggers a warning. By using inequalities, you define the 'normal' zone as a conjunction (e.g., \(120^\circ\text{F} \le T \le 180^\circ\text{F}\)) and the 'alert' zone as a disjunction (e.g., \(T < 120^\circ\text{F}\) or \(T > 180^\circ\text{F}\)). This helps automated systems prioritize safety by identifying when conditions deviate from the acceptable range.