Comprehensive Study Guide: Linear Equations and Linear Inequalities

Classifications and Rules for Linear Equations

• Classification Types:

  • Conditional Equation: An equation that is true for only specific value(s) of the variable. Example: $x + 2 = 5$ (which only works if $x = 3$).
  • Identity: An equation that is true for every possible value of the variable ($\text{Works ALL REAL NUMBERS}$). Example: $2x + 3 = 2x + 3$, which simplifies to $0 = 0$.
  • Contradiction: An equation that is never true and has no solution ($\text{NO Solution}$ / Impossible!). Example: $0 = 5$.

• General Solution Rules:

  • If an equation simplifies to a false statement such as $0 = 1$ or $0 = 12$, nothing works, meaning there is $\text{NO Solution}$ (Contradiction).
  • If an equation simplifies to an identical statement on both sides such as $0 = 0$ or $3 = 3$, everything works, meaning the solution set is $\text{ALL REAL NUMBERS}$ (Identity).

Solving Linear Equations: Worked Examples

• Homework Context:

  • Due Date: $8126\,126$

• Question 1:

  • Equation: $x + \frac{1}{2} = -3(3x - 4)$
  • intermediate expansion / record: $\frac{14}{1}x + \frac{1}{2} = -2x + 12$
  • Add $2x$ to both sides: $1x + 1 = 12$
  • Solutions / recorded steps: $x = 23$, $4$, $x = 414$

• Question 2:

  • Equation: $-5y + 5(8y - 2) = 8 + 20$
  • Step steps: $(54) + 40y - 10 = (8) + 2^2 - 24$
  • Simplify: $35y - 10 = 10 - 24$
  • Addition step: $+2y$
  • Simplify: $37y - 18 = 10$
  • Solve for $y$: $37y = 20 \rightarrow y = \frac{20}{37}$ (also recorded as $y = \frac{30}{37}$)

• Question 3:

  • Equation: $13x + 5 = -1(5x - 4)$
  • Intermediate step: $3x + 5 = -3276x + 4$
  • Recorded expressions: $\frac{7}{16}x + \sqrt{\sqrt{3}} = 4$
  • Linear step: $76x = 199$
  • Result: $x = \frac{114}{35}$

• Question 4:

  • Equation: $46 = 20 - 3(-5 - 3w)$
  • Distribute: $46 = 20 + 15 + 9w$
  • Combine constants: $46 = 35 + 9w$
  • Subtract $35$ from both sides: $11 = 9w$
  • Result: $w = \frac{11}{9}$ (or $4 = w$)

• Question 5:

  • Equation: $7(x - 5) - 2 = -4(x - 3)$
  • Distribute: $7x - 35 - 2 = -4x + 12 \rightarrow 7x - 37 = -4x + 12$
  • Add $4x$ to both sides: $11x - 37 = 12$
  • Add $37$ to both sides: $11x = 49$
  • Result: $x = \frac{49}{11}$

• Question 6:

  • Equation: $3(8x + 5) + 7(8x + 7) + 4 = 80x + 69$
  • Expand: $24x + 15 + 56x + 49 + 4 = 80x + 69 + \frac{31}{3}/2$
  • Combine like terms: $80x + 68 = 80x + 69$
  • Subtract $80x$ from both sides: $68 = 69 \rightarrow 0 + 68 = 69 \rightarrow 0 = 12$
  • Classification: Contradiction $\rightarrow \text{NO Solution}$

• Question 7:

  • Equation: $2(y - 3) + 18 = 6(6y - 1) - 19$
  • Expand and simplify: $10y + 18 = 36y - 6 - 19 \rightarrow 10y + 12 = 36y - 25$
  • Subtract $36y$ from both sides: $-26y + 12 = -25$
  • Subtract $12$ from both sides: $-26y = -37$
  • Result: $y = \frac{37}{26}$

• Question 8:

  • Equation: $4 - 20(y + 3) - 16 = -16(5y - 1) - 18$
  • Expand: $4 - 20y - 60 - 16 = -80y + 16 - 18$
  • Rearrange: $-80y + 12 - 16 = -80y + 16 - 18$
  • Simplify: $-4 = -80y - 2$
  • Addition step: $+80y$
  • Solution step: $x = 0$

• Question 9:

  • Equation simplification results in: $x = 0$ or $\text{NO Solution}$

• Question 10:

  • Equation: $3(8y + 5) - 12 = -6(-4y + 2) + 15$
  • Expand: $24y + 15 - 12 = 24y - 12 + 15$
  • Combine constants: $24y + 3 = 24y + 3$
  • Subtract $24y$ from both sides: $3 = 3 \rightarrow 0 = 0$
  • Classification: Identity $\rightarrow \text{All Real Numbers}$

Principles of Linear Inequalities and Notation

• Core Rule: Point to the Smarter # (When multiplying or dividing both sides by a negative number, reverse the direction of the inequality sign).
• Compound Statements:
  • AND: Both inequality conditions must be satisfied simultaneously. The solution set represents the intersection of the two conditions.
  • OR: At least one inequality condition must be satisfied. The solution set represents the union of the two conditions using the union symbol $\cup$.
• Interval Notation Conventions:
  • Use brackets [ or ] when an endpoint is included ($\le$ or $\ge$).
  • Use parentheses ( or ) when an endpoint is excluded ($<$ or $>$), as well as for negative or positive infinity ($-\infty$ or $\infty$).

Solving Linear Inequalities: Worked Examples

• Question 1 (Source 3):

  • System: $13x + 16 \le 8x - 9$ AND $-25 + 2x \le 9x - 4$
  • First inequality: $13x + 16 \le 8x - 9 \rightarrow 5x \le -25 \rightarrow 5x \le -10 \rightarrow x \le -2$
  • Second inequality: $-25 + 2x \le 9x - 4 \rightarrow -7x \le 21 \rightarrow x \ge -3$
  • Number Line Values: $-4, -3, -2, -1, 0$
  • Solution Set: $x \ge -3$ AND $x \le -2$
  • Interval Notation: $[-3, -2]$

• Question 2 (Source 3):

  • System: $10x - 6 > 3x + 15$ OR $12 + 8x \le 13x + 7$
  • First inequality: $10x - 6 > 3x + 15 \rightarrow 7x - 6 > 15 \rightarrow 7x > 21 \rightarrow x > 3$
  • Second inequality: $12 + 8x \le 13x + 7 \rightarrow 12 - 5x \le 7 \rightarrow -5x \le -5 \rightarrow x \ge 1$
  • Number Line Values: $0, 1, 2, 3, 4$
  • Solution Set: $x \ge 1$
  • Interval Notation: $[1, \infty)$

• Question 3 (Source 3):

  • System: $12x - 5 > 9x + 4$ OR $-8 + 4x \le 6x - 4$
  • First inequality: $12x - 5 > 9x + 4 \rightarrow 3x > 9 \rightarrow x > 3$
  • Second inequality: $-8 + 4x \le 6x - 4 \rightarrow -8 - 2x \le -4 \rightarrow -2x \le 4 \rightarrow x \ge -2$
  • Number Line Values: $-3, -2, -1, 0, 1, 2, 3$
  • Solution Set: $x \ge -2$
  • Interval Notation: $[-2, \infty)$

• Question 4 / Point to Smarter # (Source 4):

  • Steps shown: $-3x + 354 + 7 \rightarrow -7x + 373 = -35 \rightarrow -17x \le -20$
  • Additional division step: $-7 \rightarrow 400 \rightarrow -5x$
  • Number Line Values: $-4, -3, -2, -1, 0, 1, 2, 3, 4$
  • Interval Notation: $(-\infty, -3] \cup [2, \infty)$

• Question 5 (Source 4):

  • System: $12x + 32 \ge 5x + 59$ AND $2(-6x + 1) - 2 < -4x - 16$
  • First inequality: $12x + 32 \ge 5x + 59 \rightarrow 7x + 234 \ge 59 \rightarrow 7x \ge 56 \rightarrow x \ge 8$ (or $x \ge 2$)
  • Second inequality: $2(-6x + 1) - 2 < -4x - 16 \rightarrow -12x + 2 - 2 < -4x - 16 \rightarrow -12x < -4x - 16 \rightarrow -8x < -16 \rightarrow x > 2$ (or $x < 8$)
  • Number Line Values: $-1, 0, 1, 2, 3, 4, 5, 6, 7, 8$
  • Interval Notation: $[2, 8)$

• Question 6 (Source 4):

  • System: $4x + 7 < 2x - 35$ OR $2(4x + 1) - 2 \le 10x + 4$
  • First inequality: $4x + 7 < 2x - 35 \rightarrow 2x + 7 < -35 \rightarrow 7x < -42 \rightarrow x < -6$
  • Second inequality: $2(4x + 1) - 2 \le 10x + 4 \rightarrow 8x + 2 - 2 \le 10x + 4 \rightarrow -2x \le 4 \rightarrow x \ge -2$
  • Alternate line step: $6(x) - 2 \le -5x - 3 \rightarrow 6x - 24 - 2 \le -5x - 3 \rightarrow 11x - 26 \le -3 \rightarrow 11x \le 23 \rightarrow x \le \frac{23}{11}$
  • Number Line Values: $-6, -5, -4, -3, -2, -1, 0, 1, 2$
  • Interval Notation: $(-\infty, -6) \cup [-2, \infty)$

Double Inequalities and Multi-Step Systems

• Question 7 / Koh:

  • Double Inequality: $-15 < 4x - 7 \le 9$
  • Add $7$ to all three parts: $-15 + 7 < 4x \le 9 + 7 \rightarrow -8 < 4x \le 16$
  • Divide all three parts by $4$: $\frac{-8}{4} < \frac{4x}{4} \le \frac{16}{4} \rightarrow -2 < x \le 4$
  • Error Warning: Formatting as $(-\infty, -2] \cup (-\infty, 4)$ is marked wrong.
  • Correct Interval Notation: $(-2, 4]$

• Question / Rad 21:

  • System: $-4x - 7 > -5x - 3$ OR $3(2x + 5) - 9 \ge 9x + 27$
  • First inequality: $-4x - 7 > -5x - 3 \rightarrow +5x \rightarrow x - 7 > -3 \rightarrow +7 \rightarrow x > 4$ ("x 4 greater")
  • Second inequality: $3(2x + 5) - 9 \ge 9x + 27 \rightarrow 6x + 15 - 9 \ge 9x + 27 \rightarrow 6x + 6 \ge 9x + 27 \rightarrow -9x \rightarrow -3x + 6 \ge 27 \rightarrow -6 \rightarrow -3x \ge 21 \rightarrow x \le -7$
  • Number Line Values: $-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4$
  • Interval Notation: $(-\infty, -7] \cup (4, \infty)$ (Note: Must enter in Interval format to input correctly).

• Division to Part 2 / Continuous Chain:

  • Double Inequality: $-17 \le 2x - 7 < 3$
  • Add $7$ to all three parts: $-10 \le 2x < 10$
  • Divide all three parts by $2$: $-5 \le x < 5$
  • Meaning: $x \ge -5$ and $x < 5$ ("everything between").
  • Interval Notation: $[-5, 5)$