Absolute Values and Inequalities Detailed Notes

Absolute Value Definition

• The absolute value of a number is its distance from zero.
• It's always non-negative.
• |x| = x if x \geq 0
• |x| = -x if x < 0
• Example: |-3| = 3

Solving Absolute Value Equations

• If |variable| = positive_number, then the variable equals the positive number or its negative form.
• Example: If |5x + 3| = 7, then 5x + 3 = 7 or 5x + 3 = -7
  • Check solutions by plugging them back into the original equation.
  • |5(2/5) + 3| = |2 + 3| = |5| = 5 \neq 7. This appears to be an error in the original transcript. The provided 'example' does not align with prior explanation.

Isolating the Absolute Value

• If other terms are present outside the absolute value, isolate the absolute value first before breaking it apart.
• Analogous to isolating x when solving linear equations.

Absolute Value Equals Zero

• If |expression| = 0, there is only one solution. Simply remove the absolute value and solve for the variable.
• There isn't a positive or negative zero.
• Example: If |x + 2| = 0, then x + 2 = 0, so x = -2

Absolute Value Equals a Negative Number

• If |expression| = negative_number, there is no solution.
• The absolute value will always be non-negative.
• Example: |x| = -5 has no solution.

Advanced Absolute Value Equations

• For equations like |A| = B, consider two possibilities:
  • A = B
  • A = -B
• Solve both equations to find all possible solutions.
• Check for extraneous solutions.

Four Potential Cases (Reduced to Two)

• Given an equation like a = b, where either a or b may be positive or negative, there are four initial possibilities:
  1. 3x + 2 = 5 + x
  2. 3x + 2 = -(5 + x)
  3. -(3x + 2) = 5 + x
  4. -(3x + 2) = -(5 + x)
• Cases 1 and 4 are equivalent, as are Cases 2 and 3.

Absolute Value Inequalities

Understanding Distance

• |x| < 3 means all numbers within a distance of 3 from zero.
• This includes numbers between -3 and 3 (exclusive).
• |x| \leq 3 includes -3 and 3.

Solving Absolute Value Inequalities

1. Solve the corresponding equation.
2. Place the solutions on a number line.
3. Shade according to the original inequality.

LIGO (Less Inside, Greater Outside)

• LIGO is a mnemonic to remember how to shade on the number line.
• Less Than: Shade inside the two numbers.
• Greater Than: Shade outside the two numbers.

Example

• |2x - 12| < 4
  1. Solve 2x - 12 = 4 to get x = 8 and 2x - 12 = -4 to get x = 4
  2. Place 4 and 8 on the number line.
  3. Since it's "less than," shade inside between 4 and 8.
  4. Use parentheses ( ) because there's no "or equal to."
  5. Solution in interval notation: (4, 8)

Greater Than

• |x| > 3 means all numbers with a distance greater than 3 from zero.
• All numbers less than -3 or greater than 3.

Example

• |x + 7| > 3
  • Solve x + 7 = 3 to get x = -4
  • Solve x + 7 = -3 to get x = -10
  • Since it's "greater than," shade outside.
  • Interval notation: (-\infty, -4) \cup (10, \infty)

Special Cases

Absolute Value Greater Than a Negative Number

• If you have an absolute value greater than a negative number, the solution is all real numbers.
• |expression| > negative_number
• This is because the absolute value is always non-negative, thus always greater than a negative number.
• Example: |x + 3| > -2 is true for all real numbers.