Solved Exercises on Compound Inequalities in Integrated Math 1

Master compound inequalities: 'and' and 'or' inequalities through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 'And' inequality (intersection)
Exercise 1
Solve for x:
\(x > 2 \text{ and } x < 8\)
Definition:

Compound inequality: Two or more inequalities connected by 'and' or 'or'

'And' inequality: Both conditions must be true simultaneously (intersection)

Interval notation: A concise way to represent solution sets

'And' inequality method:

To solve 'and' inequalities:

  1. Solve each inequality separately
  2. Find the intersection: Values that satisfy BOTH conditions
  3. Express solution as combined inequality
  4. Write in interval notation
Original
\(x > 2 \text{ and } x < 8\)
Combine
\(2 < x < 8\)
Step 1: Analyze both conditions

\(x > 2\): x must be greater than 2

\(x < 8\): x must be less than 8

Step 2: Find intersection (values satisfying both)

x must be greater than 2 AND less than 8

This means: \(2 < x < 8\)

Step 3: Express the solution

The solution is all real numbers between 2 and 8

\(2 < x < 8\)
Final answer:

The solution is \(2 < x < 8\), or in interval notation: \((2, 8)\)

Applied rules:

• Intersection: For 'and', find values that satisfy both conditions

• Combined Inequality: Can write \(x > 2\) and \(x < 8\) as \(2 < x < 8\)

• Interval Notation: \((a, b)\) represents all numbers between a and b (not including endpoints)

Tip: 'And' means intersection - both conditions must be true!
Tip: On a number line, 'and' creates a bounded interval.
2 'Or' inequality (union)
Exercise 2
Solve for x:
\(x < -3 \text{ or } x > 5\)
Definition:

'Or' inequality: Either condition can be true (union)

Union: All values that satisfy at least one condition

Disjoint intervals: Separate ranges with no overlap

Original
\(x < -3 \text{ or } x > 5\)
Step 1: Analyze both conditions

\(x < -3\): x must be less than -3

\(x > 5\): x must be greater than 5

Step 2: Find union (values satisfying either condition)

x can be less than -3 OR greater than 5

There is no overlap between these conditions

Step 3: Express the solution

The solution includes all numbers less than -3 AND all numbers greater than 5

\(x < -3 \text{ or } x > 5\)
Final answer:

The solution is \(x < -3 \text{ or } x > 5\), or in interval notation: \((-\infty, -3) \cup (5, \infty)\)

Applied rules:

• Union: For 'or', find values that satisfy at least one condition

• Disjoint Sets: No overlap between solution intervals

• Interval Notation: Use \(\cup\) to join separate intervals

Tip: 'Or' means union - either condition can be true!
Tip: On a number line, 'or' often creates separate regions with gaps.
3 Combined inequality
Exercise 3
Solve for x:
\(-2 \leq x + 3 < 7\)
Definition:

Combined inequality: A single expression with multiple inequality symbols

Three-part inequality: Equivalent to two simultaneous conditions

Equivalent to: \(-2 \leq x + 3\) AND \(x + 3 < 7\)

Original
\(-2 \leq x + 3 < 7\)
Subtract 3 from all parts
\(-5 \leq x < 4\)
Step 1: Write the compound inequality

\(-2 \leq x + 3 < 7\)

Step 2: Subtract 3 from all three parts

\(-2 - 3 \leq x + 3 - 3 < 7 - 3\)

Step 3: Simplify all parts

\(-5 \leq x < 4\)

Step 4: Express the solution

The solution is all real numbers between -5 and 4, including -5 but not including 4

\(-5 \leq x < 4\)
Final answer:

The solution is \(-5 \leq x < 4\), or in interval notation: \([-5, 4)\)

Applied rules:

• Three-Part Inequality: Perform the same operation on all three parts

• Combined Condition: This is equivalent to \(-2 \leq x + 3\) AND \(x + 3 < 7\)

• Interval Notation: Use [ for inclusive endpoints, ( for exclusive

Tip: Treat three-part inequalities like two simultaneous conditions!
Tip: Always perform the same operation on ALL parts of the inequality!
Rules and methods, laws,...
\(a < x < b \Leftrightarrow x > a \text{ AND } x < b\)
'And' Form
\(x < a \text{ OR } x > b \Leftrightarrow x \in (-\infty, a) \cup (b, \infty)\)
'Or' Form
'And'
\(x > a \text{ and } x < b\)
Intersection: \((a, b)\)
'Or'
\(x < a \text{ or } x > b\)
Union: \((-\infty, a) \cup (b, \infty)\)
Three-part
\(a \leq x < b\)
Interval: \([a, b)\)
Mixed
\(x \geq a \text{ and } x \leq b\)
Intersection: \([a, b]\)
Compound Property: Compound inequalities connect multiple conditions.
Set Operations: 'And' means intersection, 'or' means union.
Solution: Exercises 4 to 5
4 Complex 'and' inequality
Exercise 4
Solve for x:
\(2x - 1 > 3 \text{ and } 3x + 2 \leq 14\)
Definition:

Complex compound inequality: An 'and' inequality with multi-step individual conditions

Step-by-step solving: Solve each part separately, then find intersection

Condition 1
\(2x - 1 > 3 \Rightarrow x > 2\)
Condition 2
\(3x + 2 \leq 14 \Rightarrow x \leq 4\)
Intersection
\(2 < x \leq 4\)
Step 1: Solve first inequality: \(2x - 1 > 3\)

\(2x - 1 > 3\)

\(2x > 4\)

\(x > 2\)

Step 2: Solve second inequality: \(3x + 2 \leq 14\)

\(3x + 2 \leq 14\)

\(3x \leq 12\)

\(x \leq 4\)

Step 3: Find intersection (both must be true)

\(x > 2\) AND \(x \leq 4\)

Therefore: \(2 < x \leq 4\)

Step 4: Express the solution

The solution is all real numbers greater than 2 and less than or equal to 4

\(2 < x \leq 4\)
Final answer:

The solution is \(2 < x \leq 4\), or in interval notation: \((2, 4]\)

Applied rules:

• Separate Solving: Solve each inequality independently first

• Intersection: Find values that satisfy both conditions

• Interval Notation: Use combination of parentheses and brackets as needed

Tip: Solve each part separately, then find the intersection!
5 Complex 'or' inequality
Exercise 5
Solve for x:
\(4x - 3 < 5 \text{ or } 2x + 1 > 9\)
Definition:

Complex 'or' inequality: An 'or' inequality with multi-step individual conditions

Union solving: Solve each part separately, then find union

Condition 1
\(4x - 3 < 5 \Rightarrow x < 2\)
Condition 2
\(2x + 1 > 9 \Rightarrow x > 4\)
Union
\(x < 2 \text{ or } x > 4\)
Step 1: Solve first inequality: \(4x - 3 < 5\)

\(4x - 3 < 5\)

\(4x < 8\)

\(x < 2\)

Step 2: Solve second inequality: \(2x + 1 > 9\)

\(2x + 1 > 9\)

\(2x > 8\)

\(x > 4\)

Step 3: Find union (either can be true)

\(x < 2\) OR \(x > 4\)

This represents all numbers less than 2 OR greater than 4

Step 4: Express the solution

The solution includes all numbers less than 2 and all numbers greater than 4

\(x < 2 \text{ or } x > 4\)
Final answer:

The solution is \(x < 2 \text{ or } x > 4\), or in interval notation: \((-\infty, 2) \cup (4, \infty)\)

Applied rules:

• Separate Solving: Solve each inequality independently first

• Union: Include values that satisfy either condition

• Interval Notation: Use \(\cup\) to join separate solution sets

Tip: For 'or', include solutions from both parts!
Tip: The solution may consist of separate intervals with gaps.
Comprehensive Guide to Compound Inequalities
\(\text{'And': } x > a \text{ and } x < b \Rightarrow a < x < b\)
Intersection Principle
Key definitions:

Compound inequality: Two or more inequalities joined by 'and' or 'or'

'And' inequality: Both conditions must be satisfied (intersection of sets)

'Or' inequality: Either condition can be satisfied (union of sets)

Intersection: Values that satisfy all conditions simultaneously

Union: Values that satisfy at least one condition

Complete methodology:
  1. Identify the connector: Determine if it's 'and' (intersection) or 'or' (union)
  2. Solve individually: Solve each inequality separately
  3. Find the set operation: For 'and', find intersection; for 'or', find union
  4. Express solution: Write as a combined inequality or in interval notation
Tip 1: 'And' means intersection - both conditions must be true!
Tip 2: 'Or' means union - either condition can be true!
Tip 3: For three-part inequalities, perform operations on all parts!
Tip 4: Interval notation: [ for inclusive, ( for exclusive endpoints.
Tip 5: Always verify with test values within and outside the solution set.
Common errors: Confusing 'and' with 'or', mixing up intersection/union, incorrect interval notation, forgetting to flip inequality signs.
Exam preparation: Practice both 'and' and 'or' inequalities, master interval notation, understand graphical representation.
Formulas to know by heart:

• 'And' (Intersection): \(x > a \text{ and } x < b \Rightarrow a < x < b\)

• 'Or' (Union): \(x < a \text{ or } x > b \Rightarrow x \in (-\infty, a) \cup (b, \infty)\)

• Three-part: \(a \leq x < b \Rightarrow x \in [a, b)\)

• Interval notation: \((a,b)\) for exclusive, \([a,b]\) for inclusive endpoints

• Set operations: \(\cap\) for intersection, \(\cup\) for union

Compound Inequalities Workflow

📊
Solving Process
1
Identify Connector
2
Solve Individually
3
Apply Set Operation
4
Express Solution
Compound Inequality Types
'And': Intersection (overlap) → bounded interval
'Or': Union (either) → separate intervals
Three-part: Combined condition → single interval
Interval notation: [ ] for inclusive, ( ) for exclusive
Master 'And' vs 'Or' to Excel in Algebra!

Questions & Answers

Question: How do I know if I should use 'and' or 'or' when solving compound inequalities?

Answer: The connector ('and' or 'or') is usually given in the problem, but here's how to think about them:

'And' (Intersection): Both conditions must be true simultaneously

  • Example: \(x > 2\) AND \(x < 8\) means \(2 < x < 8\)
  • Represents overlapping values
  • Creates a bounded interval

'Or' (Union): Either condition can be true

  • Example: \(x < -3\) OR \(x > 5\) means any number less than -3 OR greater than 5
  • Represents all values from both conditions
  • Often creates separate intervals

Think of 'and' as needing to satisfy both requirements, like needing both a key and a password. 'Or' is like having either a key or a password - one is sufficient.

Question: What's the difference between \((2, 5)\) and \([2, 5]\) in interval notation?

Answer: The difference is about whether the endpoints are included:

Parentheses ( ): Exclusive endpoints - the boundary values are NOT included

  • \((2, 5)\) means \(2 < x < 5\) - all numbers between 2 and 5, but not including 2 and 5
  • Represented on a number line with open circles at 2 and 5

Brackets [ ]: Inclusive endpoints - the boundary values ARE included

  • \([2, 5]\) means \(2 \leq x \leq 5\) - all numbers between 2 and 5, including 2 and 5
  • Represented on a number line with closed circles at 2 and 5

Mixed notation like \([2, 5)\) means \(2 \leq x < 5\) - includes 2 but not 5.

Remember: Parentheses look like 'open' circles, Brackets look like 'closed' circles!

Question: Can a compound inequality have no solution? What does that look like?

Answer: Yes! A compound inequality with 'and' can have no solution if the conditions cannot both be true simultaneously.

Example: \(x > 5\) AND \(x < 3\)

  • No number can be both greater than 5 AND less than 3 at the same time
  • This has no solution
  • Written as: "No solution" or using the empty set symbol \(\emptyset\)

However: A compound inequality with 'or' will almost always have a solution (except in very specific cases with contradictory conditions).

For 'or', if even one condition can be satisfied, there's a solution. For 'and', both conditions must be satisfiable simultaneously.