M217 Math 080 Unit Day 1 Notes

Topic 1: Absolute Value Inequalities

• Examples:

  • $3 |x| + 1 \leq 13$
  • 6 |x| - 1 < 15

• Steps to Solve:

  1. Isolate the absolute value expression.
  2. Create two inequalities:
  • First inequality remains as it is.
  • Second inequality involves flipping the inequality sign and changing the number’s sign.
  1. Solve each inequality separately.
  2. Combine solutions using interval notation.

• Remember:

  • Symbols: <, ≤ indicate "AND" (use barbell graph):
  • Solutions are in the format: (-#, #) or [-#, #]
  • Symbols: >, ≥ indicate "OR" (use scarecrow graph):
  • Solutions are in the format: (-∞, #) \cup (#, ∞) or (-∞, #] \cup [#, ∞)

Topic 2: Square Root Equations

• Examples:
  • $\sqrt{7x + 4} = 9$
  • $\sqrt{9x^2 - 2} = x$
• Steps to Solve:
  1. Isolate the square root on one side of the equation.
  2. Square both sides to eliminate the square root:
  • May need to use FOIL when squaring the opposite side.
  1. Rearrange the equation:
  • Move variable terms to one side and constants to the other.
  • If there is an $x^2$ term, move all terms to one side and factor.
  1. Check each solution in the original equation to ensure validity (both sides are equal).