Compound Inequalities

A compound inequality merges two separate inequalities using the words “and” or “or”. Conjunction means both must hold simultaneously, like 1 < x < 3, while disjunction uses “or,” requiring at least one condition to be true.
 

1. Solve Each Inequality Separately
For example, in the compound inequality x > 1 and x ≤ 4, treat each part as its own inequality and solve for x. 

2. Plot Each Inequality on the Same Number Line
After plotting, use open dots for strict inequalities (< or >) and closed dots for inclusive ones (≤ or ≥). Shade to the left for “<” or “≤” and to the right for “>” or “≥”.

3. Combine the Shaded Regions

• AND (conjunction): Find the overlapping section (intersection) of both shaded regions. For example, x > 1 and x ≤ 4 gives the solution 1 < x ≤ 4.
• OR (disjunction): Merge both regions (union). E.g., x < −1 or x > 2 gives the solution (−∞, −1] ∪ (2,∞).    
   

1. Recognize “AND” vs “OR”:

• AND → solution is the overlap (intersection)
• OR → solution is the combined regions (union)

2. Break the compound inequality into two separate inequalities. Solve each part as usual:
We need to isolate x while solving an inequality. However, when multiplying and dividing both sides by a negative number, we should flip the inequality sign.

3. Combine solutions:

• AND → take the intersection.
• OR → take a union.

4. (Optional) Graph on a number line to verify the solution.
 

Compound inequalities are used in our daily life. It helps us in budgeting and monitoring speed limits and temperature ranges. We will be learning in the field of architecture, nature, biology, art, and design also. 

• Biology: Enzyme Activity Range - Biologists determine the temperature and pH ranges at which enzymes function optimally. For instance, an enzyme may function only when the pH is between 6 and 8. This can be modeled with an  “and” compound inequality.

• Architecture: Structures such as the Tower of Light, inspired by mollusc shells, rely on compound inequalities: the material stress must not exceed its yield strength, and the deflection must stay within acceptable limits, ensuring safety and integrity.

• Art & Design: The golden ratio sets minimum and maximum bounds on proportions. The ratio between different parts of the Mona Lisa must lie between approximately 1.618 and its reciprocal, ensuring balanced composition.

• Nature: Fractal Branching for Resource Transport, Tree and blood vessel systems follow fractal designs that balance branching depth (to reach all areas) without exceeding flow resistance, maintaining optimal transport efficiency.

• Cross-disciplinary (Nature ↔ Built Environment): Hexagonal Honeycomb Efficiency in Architecture, Designs like the Eden Project use hexagonal structures that must fit within minimum material thickness while supporting maximum load, combining two inequality constraints for strength and lightness