And vs. Or

1. Fundamental Concepts

• A compound inequality is a mathematical statement that combines two inequalities using the words “and” or “or.”

• It describes a range (or multiple ranges) of possible values for a variable.

• There are two main types:

  • “And” compound inequalities: solutions that satisfy both inequalities.

  • “Or” compound inequalities: solutions that satisfy at least one of the inequalities.

2. Key Concepts

A. “And” Compound Inequality

• Often written in the form:
  $a < x < b$a or $a \leq x \leq b$a≤x≤b
  This means x must be greater than a and less than b.

• The solution is the overlap (intersection) of both inequalities.

B. “Or” Compound Inequality

• Written as two separate inequalities joined by “or”:
  $x < a \text{ or } x > b$xb

• The solution includes all values that satisfy either condition.

• The graph of the solution often has two separate regions.

C. Solving Compound Inequalities

• Solve each inequality individually.

• For “and” inequalities written in chain form (e.g., $a < 2x + 1 \leq b$a<2x+1≤b), isolate the variable by performing the same operations across all parts.

• For “or” inequalities, solve each side and combine the solutions.

D. Graphical Representation

• Use a number line to visually represent the solution.

• “And” is a connected region (interval), “or” is often two disjoint regions.

3. Examples (Low, Medium, and High Difficulty)

🟢 Low-Level Example (Basic ‘and’ compound inequality):

Solve: $2 < x \leq 5$2
Solution: All values of x greater than 2 and less than or equal to 5.

🟡 Medium-Level Example (Two-step solving ‘and’):

Solve: $-3 \leq 2x - 1 < 5$−3≤2x−1<5
Step 1: Add 1 to all parts: $-2 \leq 2x < 6$−2≤2x<6
Step 2: Divide all parts by 2: $-1 \leq x < 3$−1≤x<3
Solution: $x \in [-1, 3)$x∈[−1,3)

🔴 High-Level Example (‘or’ compound inequality with solution set union):

Solve: $3x - 5 < -2 \text{ or } 2x + 1 > 7$3x−5<−2 or 2x+1>7
First inequality:
$3x - 5 < -2 \Rightarrow x < 1$3x−5<−2⇒x<1
Second inequality:
$2x + 1 > 7 \Rightarrow x > 3$2x+1>7⇒x>3
Solution: $x < 1 \text{ or } x > 3$x<1 or x>3

4. Problem-Solving Techniques

• Identify the type of compound inequality: “and” or “or”.

• Solve each part separately; for chained “and” statements, treat it as one inequality but apply the same operations to all three sides.

• Isolate the variable by using inverse operations.

• Graph the solution to visualize the answer—this helps with interpreting the range of values.

• Write the solution in correct interval notation or inequality form.

• Check endpoints for inclusion (use ≤/≥ or open/closed circles on graphs).

• Watch for contradictions (e.g., $x < 2 \text{ and } x > 5$x<2 and x>5 → no solution).