Notes on Absolute Value, Equations, Inequalities, and Cartesian Tools

Absolute Value: Key Concepts

• Absolute value is the distance from 0 on the real number line; $|a| \ge 0.$
• Basic properties: $|-a| = |a|,\quad |ab| = |a||b|,\quad \left|\frac{a}{b}\right| = \frac{|a|}{|b|},\quad b \neq 0.$
• Interpretation: $|x| = a \Rightarrow x = -a \text{ or } x = a.$
• Distance on the number line: $|x - y|.$
• Extension to the plane: distance formula and midpoint concept apply in 2D; derived from the Pythagorean theorem.

Absolute Value Equations

• Equations with absolute value symbols: solve by splitting into two cases where the inside equals ± the value. Be aware of extraneous solutions.
• Rule: if $|f(x)| = k$, then solve $f(x) = k$ or $f(x) = -k.$ Check both in the original equation.
• Example 1: $|x+5| = 7 \Rightarrow x+5 = 7 \text{ or } x+5 = -7 \Rightarrow x = 2, -12.$
• Example 2: $|1+2x| = 5 \Rightarrow 1+2x = 5 \text{ or } 1+2x = -5 \Rightarrow x = 2, -3.$
• Note: extraneous solutions can occur if steps introduce solutions that do not satisfy the original equation.

Absolute Value Inequalities

• Key rules (for all algebraic expressions):
  • |x| < a \iff -a < x < a.
  • $|x| \le a \iff -a \le x \le a.$
  • |x| > a \iff x < -a \text{ or } x > a.
  • $|x| \ge a \iff x \le -a \text{ or } x \ge a.$
• Note: "Less than" is an AND statement; "greater than" is an OR statement.

Applications Involving Linear Equations

• Procedure for solving word problems (pg. 93):
  1. Identify the question.
  2. Make notes.
  3. Assign a variable.
  4. Set up an equation.
  5. Solve the equation.
  6. Check the solution.
• Common application types: distance-rate-time, simple and multi-step interest, mixture problems, geometry problems, and basic algebra word problems.
• Modeling tips: estimate a practical solution before solving; check results; use intuition to verify investment and mixture problems.

Cartesian Plane: Distance and Midpoint

• Cartesian plane basics: origin $O = (0,0)$; axes: $X$-axis, $Y$-axis; four quadrants.
• Distance formula (distance between points):
  $d = \sqrt{(x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2}.$
• Pythagorean theorem: $a^2 + b^2 = c^2.$
• Midpoint formula (segment joining two points):
  $\left(x<em>m, y</em>m\right) = \left(\frac{x<em>1 + x</em>2}{2}, \frac{y<em>1 + y</em>2}{2}\right).$
• The distance formula can be viewed as a 2D extension of the distance on a number line.

Quick Reference: Modeling and Problem-Solving Steps

• Always connect problems to a linear model when appropriate; use the 6-step process above.
• Distinguish when you’re working in 1D (number line) vs 2D (plane).
• Use the distance and midpoint formulas to locate and measure points on the plane.
• Check units and reasonableness; verify answers with intuition or rough estimation.