Q: What does 'or' mean in compound inequalities?

In compound inequalities, 'OR' means at least one condition must be satisfied. It represents the union of solution sets. The solution includes values that make either inequality (or both) true. Example: x ≤ -3 OR x ≥ 2 means x is less than or equal to -3 OR greater than or equal to 2. Solution: (-∞, -3] ∪ [2, ∞). On a number line, shade both regions separately. OR creates larger solution sets than AND.

Q: How to graph compound inequalities on a number line?

To graph compound inequalities: 1) Draw a number line with key points marked. 2) For each inequality endpoint: Use open circle (○) for , closed circle (●) for ≤ or ≥. 3) For AND: Shade the overlapping region between endpoints. 4) For OR: Shade all regions satisfying either inequality. Example AND (2 4): Open circles at 1 and 4, shade left of 1 and right of 4.

Q: What is intersection vs union in compound inequalities?

Intersection (AND, symbol ∩): Values in both solution sets. Results in narrower solution. Example: x > 2 AND x 8 → union is (-∞, 2) ∪ (8, ∞). Intersection requires all conditions, union requires any condition. AND uses intersection, OR uses union. These are set operations from set theory applied to solution sets.

Question 1

What is a compound inequality?

Accepted Answer

A compound inequality is a statement that combines two or more inequalities using 'AND' or 'OR'. AND inequalities (conjunction) require both conditions to be true simultaneously, like -3 < x < 5. OR inequalities (disjunction) require at least one condition to be true, like x < -2 OR x > 3. Compound inequalities are used to describe ranges or sets of values that satisfy multiple conditions.

Question 2

How do you solve compound inequalities?

Accepted Answer

To solve compound inequalities: For AND: 1) Solve each inequality separately. 2) Find the intersection (overlap) of solutions. 3) Write in compact form if possible (a < x < b). For OR: 1) Solve each inequality separately. 2) Find the union (combination) of solutions. 3) Write with ∪ symbol. Example AND: x > 2 AND x < 5 → solution: 2 < x < 5 or (2, 5). Example OR: x < 1 OR x > 4 → solution: (-∞, 1) ∪ (4, ∞).

Question 3

What is the difference between AND and OR compound inequalities?

Accepted Answer

AND (Conjunction): Both conditions must be true. Represents intersection. Solutions satisfy all inequalities. Typically written as a < x < b for continuous ranges. Example: x > 3 AND x < 7 means x is between 3 and 7. OR (Disjunction): At least one condition must be true. Represents union. Solutions satisfy one or more inequalities. Uses ∪ symbol in interval notation. Example: x < 2 OR x > 5 means x is less than 2 or greater than 5. AND narrows the solution, OR expands it.

Question 4

How to write compound inequalities in interval notation?

Accepted Answer

Interval notation for compound inequalities: AND (continuous range): Use single interval. Example: -2 ≤ x ≤ 5 becomes [-2, 5]. Example: 3 < x < 7 becomes (3, 7). OR (separate ranges): Use union symbol ∪. Example: x < -1 OR x > 2 becomes (-∞, -1) ∪ (2, ∞). Brackets: [ ] for ≤ or ≥ (closed/included), ( ) for < or > (open/excluded). Infinity always uses ( ).

Question 5

What does 'and' mean in compound inequalities?

Accepted Answer

In compound inequalities, 'AND' means both conditions must be satisfied simultaneously. It represents the intersection of solution sets. The solution includes only values that make both inequalities true. Example: x > -2 AND x ≤ 3 means x must be greater than -2 AND also less than or equal to 3. Solution: -2 < x ≤ 3 or (-2, 3]. On a number line, this is the overlapping region. If there's no overlap, there's no solution (empty set).

Question 6

What does 'or' mean in compound inequalities?

Accepted Answer

In compound inequalities, 'OR' means at least one condition must be satisfied. It represents the union of solution sets. The solution includes values that make either inequality (or both) true. Example: x ≤ -3 OR x ≥ 2 means x is less than or equal to -3 OR greater than or equal to 2. Solution: (-∞, -3] ∪ [2, ∞). On a number line, shade both regions separately. OR creates larger solution sets than AND.

Question 7

How to graph compound inequalities on a number line?

Accepted Answer

To graph compound inequalities: 1) Draw a number line with key points marked. 2) For each inequality endpoint: Use open circle (○) for < or >, closed circle (●) for ≤ or ≥. 3) For AND: Shade the overlapping region between endpoints. 4) For OR: Shade all regions satisfying either inequality. Example AND (2 < x < 5): Open circles at 2 and 5, shade between. Example OR (x < 1 OR x > 4): Open circles at 1 and 4, shade left of 1 and right of 4.

Question 8

Can a compound inequality have no solution?

Accepted Answer

Yes, compound inequalities can have no solution (empty set ∅). This happens with AND inequalities when the conditions contradict: Example: x > 5 AND x < 3 has no solution (no number is simultaneously greater than 5 and less than 3). Example: x > 10 AND x < 8 → ∅ (impossible). OR inequalities rarely have no solution—they only would if both individual inequalities are impossible. Solution set notation: ∅ or { } for empty set. Always check if AND conditions can overlap.

Question 9

What is intersection vs union in compound inequalities?

Accepted Answer

Intersection (AND, symbol ∩): Values in both solution sets. Results in narrower solution. Example: x > 2 AND x < 8 → intersection is (2, 8). Union (OR, symbol ∪): Values in either solution set. Results in broader solution. Example: x < 2 OR x > 8 → union is (-∞, 2) ∪ (8, ∞). Intersection requires all conditions, union requires any condition. AND uses intersection, OR uses union. These are set operations from set theory applied to solution sets.

Question 10

How are compound inequalities used in real life?

Accepted Answer

Compound inequalities appear in: 1) Age restrictions - must be ≥18 AND <65 for certain jobs. 2) Temperature ranges - thermostat keeps temp between 68°F and 72°F (AND). 3) Health metrics - BMI outside normal range (too low OR too high). 4) Speed limits - speed must be >0 AND ≤65 mph. 5) Grade requirements - need ≥90 for A OR ≤60 for F. 6) Business - profit must be >costs AND

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