Inequalities Lecture Notes

Inequalities and Interval Notation

• 25000 > 2.0000000001 is technically true.
• Focus on values greater than a specific number.
• When learning inequalities, open or closed circles were likely used to represent the boundary.
• In college algebra, interval notation is used.
  • Open Circle: Represented with a parenthesis.
  • Closed Circle: Represented with a bracket.
• Interval notation indicates where an interval starts and stops, reading from left to right on the number line.

Expressing Intervals

• Example: Interval starting at 2 and going to infinity.
  • Starts at 2: (2,…). Parenthesis indicates that 2 is not included.
  • Ends at infinity: (…, \infty).
• Set notation: {{x | …}}, where x represents all numbers that meet certain criteria.

Number Line Representation

• Example: x \geq 2.
  • On a number line, shade to the right of 2, including 2.
  • Interval notation: [2, \infty).

More Examples

• Shading to the left of -1 (not inclusive).
  • Parenthesis on -1 because it is not included: (-1).
  • Interval notation: (-\infty, -1).
• Brackets mean "or equal to", represented by a solid line.
• Less than indicates everything to the left on the number line.

Compound Inequalities

• Example: 0.5 < x \leq 3.
  • x is between 0.5 and 3.
  • 0. 5 is not included (parenthesis), but 3 is (bracket).
  • Interval notation: (0.5, 3].

Linear Inequalities

• Solve linear inequalities similarly to equations.
• Verify the graph is correct by starting from the left.
  • Example: Graph going to the left from -4 is represented as (-\infty, -4).

Special Case: Multiplying or Dividing by a Negative

• 10 > -4 is true.
• If you multiply by a negative, you need to flip the inequality symbol.
• If you divide by a negative, also flip the inequality symbol.

Example:

• Multiplying both sides of 10 > -4 by -2 gives -20 < 8 (inequality flipped).
• Dividing both sides of 10 > -4 by -2 gives -5 < 2 (inequality flipped).
• Helpful Hint: When multiplying or dividing by a negative number, reverse the direction of the inequality symbol.

Example with Fractions

• (\frac{1}{4})x \leq 3. Multiply both sides by 4 (or \frac{4}{1}) to eliminate the fraction.
• Result: x \leq 12.
• Since we multiplied by a positive number, the inequality symbol remains the same.
• On a number line, shade everything to the left of \frac{3}{2}, with a bracket at \frac{3}{2}.
• Interval notation: (-\infty, \frac{3}{2}].

Solving Linear Inequalities: Multi-Step

• Example: 6x + 5 - x \leq 7x - 15.
• Simplify both sides: 6x - x + 5 \leq 7x - 15 becomes 5x + 5 \leq 7x - 15.
• Move variables to the left: Subtract 7x from both sides, resulting in -2x + 5 \leq -15.
• Move constants to the right: Subtract 5 from both sides, resulting in -2x \leq -20.
• Divide by -2 (and flip the inequality): x \geq 10.

Solving Linear Inequalities: Fractions

• Example: (\frac{2}{5})(X - 6) < X - 1.
• Multiply every term by 5 to eliminate the fraction: 2(X - 6) < 5X - 5.
• Distribute: 2X - 12 < 5X - 5.

Special Cases: All Real Numbers or No Solution

• If you solve an inequality and the variable disappears, you have two possibilities:
  • If the resulting statement is true, the solution is all real numbers. Interval notation: (-\infty, \infty).
  • If the resulting statement is false, there is no solution. Represented by {\emptyset}.

Compound Inequalities

• Combine two inequalities with "and" or "or".
• Example (playground age): To get on the playground you need to be greater than two years old, but less than eight ( 2 < age < 8 ).

Solving "And" Inequalities

• Solve inequality with "and" by performing operations on all parts of the inequality.

Example

• Solve -3 \leq 4 + x < 2:
  • Subtract 4: -7 \leq x < -2.
  • Interval Notation: [-7, -2).

Solving "Or" Inequalities

• Solve each inequality separately.

Example

• x + 1 \geq 4 or 5x < 6.
• Solve x + 1 \geq 4 \rightarrow x \geq 3 \rightarrow [3, \infty)
• Solve 5x < 6 \rightarrow x < \frac{6}{5} \rightarrow (-\infty, \frac{6}{5})
• Combine the intervals with a union symbol: (-\infty, \frac{6}{5}) \cup [3, \infty).