Recording-2025-07-07T15:35:10.658Z

Course Logistics & Expectations

• Documents Provided in Class
  • Blank copy of “Test 1 Notes” + fully-worked solution packet.
  • Several supplemental worksheets (each targets specific textbook sections) with answers appended to the last page of the PDF.
• Homework Philosophy
  • Instructor assigns the bare minimum (≈ 10–12 questions per section).
  • Students needing extra practice can:
  • Re-work any question via “Similar Question” (≈ 100 regenerated versions per prompt).
  • Download/print the optional worksheets.
• Hybrid (Attendance) Assignments
  • Three hybrid assignments, each due Sunday @ 11:59 PM.
  • For every test, the review sheet is the hybrid assignment (no extra tasks).
  • Acceptable submission formats: clear photos from phone, computer, iPad, etc.
    ✗ Do NOT upload .HEIC files (blocked by the LMS).
  • Upload window opens immediately; early submissions encouraged—ideal completion time: the Thursday before the due date.
  • Hybrid grade simultaneously records attendance.
  • Submitting only final answers ⇒ automatic 0 (work must be shown).
• Online HW Platform Details
  • Due Sunday @ 11:59 PM (separate from hybrid).
  • 3 attempts per item → warning after 1st & 2nd submission; 3rd reveals the answer.
  • “Help Me Solve This” walks through solution but counts the item wrong; use “View an Example” instead.
  • Unlimited practice via “Similar Question”.
  • Late work allowed with 10 % deduction (better than a 0).
• General Advice
  • Complete each night’s homework before the following class so questions can be resolved in real time.
  • Weekends: instructor’s response time slower.

ESRBC Algorithm for Solving Equations

Bullet acronym to decide what to do first:

1. E – Eliminate fractions (clear denominators).
2. S – Simplify
   • Distribute through parentheses.
   • Combine like terms on the same side.
3. R – Rearrange (use +/− to move terms across the equal sign).
4. B – ??? (The transcript never defines B; context implies “Balance” but step is merged with R in practice.)
5. D – Divide (isolate the variable via ÷ or × inverse).
6. C – Check (optional unless the problem type mandates it).

Linear-Equation Examples

Example 1

Equation: 7x-5=72
ESRBC steps

• R: add 5 → 7x=77
• D: divide by 7 → x=11
• C (optional): LHS = RHS.

Example 2

Equation: 2x-7 = x+6

• R: subtract x from right → x-7=6
• R again: add 7 → x=13
• D: nothing left to divide; solved.

Example 3 (Parentheses Both Sides)

3(x-2)+7 = 2(x+5)

• S: distribute → 3x-6+7 = 2x+10
• S: combine like terms (-6+7) ⇒ 3x+1 = 2x+10
• R: subtract 2x → x+1=10
• R: subtract 1 → x=9

Example 4 (Clearing Fractions)

\frac{x}{3}-\frac{x}{2}=2

• E: LCD = 6. Multiply every term by 6.
  6\left(\frac{x}{3}\right)-6\left(\frac{x}{2}\right)=6\cdot2
  Simplify → 2x-3x=12
• S: -x=12
• D: divide by -1 → x=-12

Example 5 (Binomial Denominator)

\frac{x+1}{3}-\frac{x-2}{7}=2

• E: LCD = 21. Multiply all three terms by 21.
  • 21\left(\frac{x+1}{3}\right) → 7(x+1)
  • 21\left(\frac{x-2}{7}\right) → 3(x-2)
  • RHS: 21\cdot2=42
• S: distribute → 7x+7 - 3x +6 =42
• S: combine → 4x+13 =42
• R: subtract 13 → 4x=29
• D: x=\frac{29}{4} (improper fraction; calculator shortcut: 2nd ABC on TI-30Xs returns \frac{29}{4}).

Special Cases (Identity & Inconsistent)

• If all variables cancel while rearranging:
  • Remaining true statement (e.g., 9=9) ⇒ Identity → solution set = all real numbers (ℝ).
  • Remaining false statement (e.g., 6=7) ⇒ Inconsistent → no solution (∅).

Identity Sample

5x+9 = 5x+9 → variables cancel, 9=9 (true) ⇒ ℝ.

Inconsistent Sample

5x+6 = 5x+7 → variables cancel, 6=7 (false) ⇒ no solution.

Rational Equations (Variable in Denominator)

Extra steps: find LCD and domain restrictions.

Workflow

1. Factor every denominator if possible.
2. Determine LCD (include each unique factor once).
3. Find Restrictions: set each LCD factor to 0, solve, and label “x\neq …”.
   • Must verify final solution ≠ any restriction.
4. Multiply entire equation by LCD → clear denominators.
5. Solve resulting linear equation via ESRBC.

Rational Example 1

\frac{4}{x} + 2 = 5 + \frac{6}{x}

• Denominators: x
• LCD: x
• Restriction: x\neq 0
• Multiply by x → 4 + 2x = 5x +6
• Solve: subtract 5x → 4 + 2x - 5x =6 ⇒