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.