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.

Study Strategy: Think of compound inequality as restrictions. Like if you are allowed to have less than three chocolate per day, then it can be said the number of chocolate you eat per day can be more than 0 but should be less than 3.

This can be mathematically expressed as 0 < x < 3, where 'x' represents the number of chocolates you eat per day.
 

To represent a compound inequality on a graph, follow the steps mentioned below:

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,∞).    

Let's practice this using a word problem

Practice Problem: Suppose you have $4, and you want to buy a toy. So, the money you can spend can be more than $0 or less than $4. Express these inequalities mathematically and on graph. 

Solution: Lets represents the money with variable x.

• For x > 0, we will use open dot at 0.
• For \( x \le 4\), we will use close dot at 4. 

Expressing these inequalities on graph:

The intersection of both regions will be the solution of \(0 < x \le 4\).

For solving compound inequalities, students needs to follow the steps given below:

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(if needed).
    
3. 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.
    
4. Combine solutions:
   AND → take the intersection.
   OR → take a union.
    
5. (Optional) Graph on a number line to verify the solution.

Let's practice these steps using problems

Practice Problem 1: Solve -2 < 2x + 4 < 3

Explanation: Given Inequality: -2 < 2x + 4 < 3

1. Subtract 4 on both sides
   \(-2 - 4 < 2x + 4 - 4 < 3 - 4\\ -6 < 2x < -1 \)
    
2. Divide by 2
   \(\frac{-6}{2} < \frac{2x}{2} < \frac{-1}{2} \\ -3 < x < \frac{-1}{2}\)

The solution is \(x = (-3, -\frac{1}{2})\) or \(x = (-\infty , - \frac{1}{2}) \cap (-3, \infty)\)

Practice Problem 2: Solve \(3 > 3x - 2 \ or\ -\frac{x}{2} + 4 < 3\)

Explanation: Given Inequalities: \(3 > 3x - 2 \ or\ -\frac{x}{2} + 4 < 3\)

• Solving -3 > 3x - 2
   

1. Add 2 on both sides
   \(-3 + 2> 3x - 2 + 2\\ -1 > 3x\)
    
2. Divide by 3
   \(\frac{-1}{3} > \frac{3x}{3} \\ \frac{-1}{3} > x \)

• Solving \(​​​​- \frac{x}{2} + 4 < 3\)
   

1. Subtracting 4 in both sides:
   \(-\frac{x}{2} + 4 - 4 < 3 - 4 \\ -\frac{x}{2} < -1\)
    
2. Multiplying by -2 on both sides
   \(-2 \times -\frac{x}{2} < -1 \times -2 \\ x > 2\)
    
3. Combining both inequality
   \( -\frac{1}{3} > x > 2\)

The solution is \(x = (- \infty, \frac{-1}{3})\cup (2, \infty)\)

Students often find compound inequalities difficult and confusion at first.  To ease the process, let’s focus on some tips and tricks to help you easily grasp the concept of compound inequalities.

1. Always make the changes on both sides of the inequality.
2. Remember, addition and subtraction on both sides of the inequality, doesn't change the sign.
3. Always reverse the inequality sign when multiplying or dividing by a negative number.
4. For x > a, shade the area outside 'a'.
5. For x < a, shade the area inside 'a'. 

Parent Tip: 

• Help children to correctly draw a shaded region on a graph.
• You can use real-life conditions for explaining inequalities efficiently.
• Encourage to solve problems from compound inequality's worksheet.

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. 

1. Biology: 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.
    
2. 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.
    
3. 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.
    
4. 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.
    
5. 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