Solving Equations and Inequalities

Exponents vs. Algebraic Terms

  • Exponents: Only the powers.
  • Algebraic Term: Expression including variables and coefficients.
  • Algebraic Expression: Combination of algebraic terms.

Solving First Degree Equations

  • Focus: Equations where the highest power of the variable is one.
  • Form: Equations like ax = b, where:
    • a and b can be integers, decimals, or fractions.
    • Solution: x = /a

Avoiding the Intuitive Approach

  • Discouraged: Directly guessing the value of x (e.g., what number multiplied by 3 equals 9).
  • Justification: Inefficient for complex numbers (e.g., 7.11x = 2.9).

General Solution

  • Equation: 7.11x = 2.9
  • Solution: x = {2.9}{7.11} (Leave it in this form; no simplification needed).

Identifying a and b

  • Importance: Recognizing a and b in the context of ax = b
  • Example: For 3x = 9, a = 3 and b = 9.
  • Note: Distinguish between identifying a and b and finding the solution for x.

Extracting Information

  • Problem: 7x = 12
    • Solution: x = {12}{7}
    • Identify: a = 7, b = 12.

Complex First Degree Equations

  • Definition: Equations not immediately in the form ax = b.
  • Significance: Require simplification before solving for x.
  • Solution: Requires Al Khourizmi’s three steps.

Al Khourizmi's Three Steps

  • Step 1: Eliminate parentheses using the distributive property.
    • a(x ± y) = ax ± ay
  • Step 2: Simplify and rearrange the equation.
    • Every time you move a term you change its sign.
  • Step 3: Isolate the variable to solve for x.

Commutative Property

  • Definition: The order of terms does not affect the result in addition and multiplication.
    • Addition: 2 + 5 = 1 + 4 = 4 + 1
    • Multiplication: 2 × 8 = 8 × 2, x × y = y × x
  • Limitation: Does not apply to subtraction (e.g., 7 - 2 ≠ 2 - 7).

Example Problem

  • Equation: 3(x + 1) - 7x + 20 = 10x - 1
  • Step 1: Distribute.
    • 3x + 3 - 7x + 20 = 10x - 1
  • Step 2: Combine like terms and rearrange.
    • Move 10x to the left: -15x
  • Step 3: Solve for x.
    • x = {3}{19}

Vertical Calculation

  • Avoid: Unnecessary multiple equal signs on one line.

Another Example

  • Equation: 3(-2x + 4) - 23x = 2(x + 1) - 5(3x)
  • Apply distributive property and simplify.
    • -6x + 12 - 23x = 2x + 2 - 15x
  • Combine like terms.
    • -29x + 12 = -13x + 2
  • Rearrange terms.
    • -16x = -10
  • Solve for x.
    • x = {30}{16} = {15}{8}
  • Identify: a = -16, b = -30.

Course Information

  • Content: Assignments, practices, and online quizzes.
  • Quizzes: 5% of the grade.
  • Attendance: 5% of the grade.
  • Projects/Assignments: Unit-based, submitted for credit.
  • Practices: Additional problems without provided answers (solutions uploaded later).
  • Assignments: Assigned on weekends, submission required.
  • Project Time Management: Work on assignments in multiple sessions to enhance understanding.

Textbook

  • Access: Via provided hyperlink.
  • Content: Solving linear equations, inequalities, and absolute values.

Screen Activation Example

  • Capacitive Sensors: Create an electric field when touched.
  • Screen Activation Model: Involves an equation owned by the company (patent).
  • Problem: If a capacitor burns out, what value c is needed to replace it?
    • Solve the relevant first-degree equation to find the capacitor value.

Indian Numbers

  • Historical Context: Origin and evolution of numerical symbols.
  • Angle-Based: Early numbers based on angles (e.g., 1 had one angle).

Inequalities

  • Definition: Two expressions related by inequality signs (
  • Single Inequality: One inequality in the expression.
  • Difference: Main difference from equality is the physical interpretations of the solutions.
  • Division by Negative: When dividing by a negative number, flip the inequality sign.

Cost Reduction

  • Design Consideration: Reducing the cost of components (e.g., using integers instead of fractions).
  • Practical Implication: Using numbers that are cost effective to implement.

Al Khourizmi and Cyphers

  • Historical Significance: Cypher originally meant zero, symbolizing the cancellation of values.

Double Inequalities

  • Definition: Two inequalities joined by logical operations (and, or).

Logical Operations: AND

  • Mathematical Symbol: Inverted U (\cap).
  • Definition: Intersection, common elements between sets.
  • Graphical Representation: Common area between regions.

Solving Joint Inequalities

  • Step 1: Solve each inequality separately.
  • Step 2: Graph each solution on a number line.
  • Step 3: Find the impact of AND/OR (intersection/union).

Example with AND

  • Inequalities: Two separate inequalities connected by “and”.
  • Solution: Find the region where the solutions overlap.
  • No Overlap: Indicates no solution (the systems will never synchronize).

Logical Operations: OR

  • Definition: Union, all elements from both sets.
  • Symbol: U.
  • Graphical Representation: All areas covered by either region.

Solving Joint Inequalities with OR

  • Solution: Union of the individual solutions.
  • If this breaks, I will know when. If this breaks, I will know when. Step equal number.

Procedure Outline

  • Step 1: Solve each inequality.
  • Step 2: Graph each inequality.
  • Step 3: Apply AND/OR operation.

Absolute Value Equations

  • Definition: The distance of a number from zero (always non-negative).
  • Notation: |x|
  • Cases: Either the value equals a number, the region less than the number, or the region greater than the number.

Solving Absolute Value

  • Case: Equal
  • Example: |5x + 2| = 20
  • Setup: Set up two equations for case equal, one positive, one negative.
  • Solutions:
    • 5x + 2 = 20 and 5x + 2 = -20

Two-Times Examples

  • 50% of problem: The setup is 50% of problem.
  • 2|6x + 1| = 5
  • 2(6x + 1) = 5 and 2(6x + 1) = -5

Recommendations

  • Writing notes, instead of pictures, is recommended unless its in class, you can’t record a video.