Tasks 02-01 - Linear Equations & Inequalities

Section 02: Equations & Systems

Problem 1: Solving Linear Inequalities (x)

Solve the following inequalities and express the solution set using interval notation.

  1. \(5x + 3 \leq 18\)

  2. \(21 - 2x > 9\)

  3. \(3(x + 4) \geq 5x - 8\)

  4. \(\frac{x-1}{4} < \frac{x+3}{2}\)

  1. \(5x \leq 15 \implies x \leq 3\). Interval: \((-\infty, 3]\)

  2. \(-2x > -12 \implies x < 6\) (sign flips). Interval: \((-\infty, 6)\)

  3. \(3x + 12 \geq 5x - 8 \implies 20 \geq 2x \implies 10 \geq x\). Interval: \((-\infty, 10]\)

  4. Multiply by 4: \(x - 1 < 2(x + 3) \implies x - 1 < 2x + 6 \implies -7 < x\). Interval: \((-7, \infty)\)

Problem 2: Break-Even Analysis (xx)

A company is launching a new smartwatch with the following cost structure:

  • Fixed monthly costs (rent, salaries, insurance): $12,000
  • Variable cost per watch (materials, assembly): $85
  • Planned selling price: $249

  1. Write an equation for the total cost when x watches are produced in the first month.

  2. How many watches must be sold in the first month to break even on monthly operations?

Using IDEA Method:

  1. Identify: Need total cost equation for x watches in first month
    • Develop: Total Cost = Fixed costs + Variable costs
    • Execute: Total Cost = 12,000 + 85x
    • Assess: Equation shows $12,000 base cost plus $85 per unit
  2. Identify: Find x where Revenue = Monthly Cost
    • Develop: Revenue = 249x, Total Cost = 12,000 + 85x
    • Execute:
      • 249x = 12,000 + 85x
      • 164x = 12,000
      • x = 12,000/164 ≈ 73.17
    • Assess: Need to sell 74 watches (round up for break-even!)

Problem 3: Motion and Meeting Problem (xxx)

Two delivery services are coordinating a package handoff.

  • QuickShip leaves City A heading east toward City B at 65 km/h.
  • At the same time, FastTrack leaves City B heading west toward City A at 85 km/h.
  • The cities are 375 km apart.

  1. How long until the drivers meet?

  2. How far from City A will they meet?

  3. If QuickShip had left 30 minutes earlier, where would they meet?

  1. Meeting time:
    • Let t = time in hours until meeting
    • Distance by QuickShip: 65t
    • Distance by FastTrack: 85t
    • Total distance: 65t + 85t = 375
    • 150t = 375
    • t = 2.5 hours
  2. Distance from City A:
    • QuickShip travels: 65 × 2.5 = 162.5 km from City A
    • Verification: FastTrack travels 85 × 2.5 = 212.5 km
    • Total: 162.5 + 212.5 = 375 km
  3. With 30-minute head start:
    • QuickShip travels alone for 0.5 hours: 65 × 0.5 = 32.5 km
    • Remaining distance: 375 - 32.5 = 342.5 km
    • Let t = time after FastTrack starts
    • 65t + 85t = 342.5
    • 150t = 342.5
    • t = 2.283 hours
    • Distance from City A: 32.5 + 65(2.283) = 32.5 + 148.4 = 180.9 km

Problem 4: Business Decision (xx)

A freelance designer has two pricing plans for a project:

  • Plan A: A flat fee of €1,200.
  • Plan B: An initial fee of €500 plus €35 per hour.

Let \(h\) be the number of hours the project takes.

  1. Write an expression for the total cost of each plan (e.g., Cost A, Cost B).
  2. For what number of hours are the two plans equal in cost?
  3. If the designer estimates the project will take 25 hours, which plan is cheaper for the client?

  1. Cost A = \(1200\). Cost B = \(500 + 35h\).

  2. Set costs equal: \(1200 = 500 + 35h \implies 700 = 35h \implies h = 20\). The plans cost the same at 20 hours.

  3. At 25 hours:

    • Cost A = 1200
    • Cost B = \(500 + 35(25) = 500 + 875 = 1375\) Plan A is cheaper for the client if the project takes 25 hours.

Problem 5: Investment Threshold (xx)

An investor has €20,000 to invest. They decide to put some money into a safe bond that yields 3% annual interest and the rest into a riskier stock fund that is projected to yield 8% annual interest.

What is the minimum amount of money the investor must put into the stock fund to ensure a total annual return of at least €1,000?

Let \(s\) be the amount invested in the stock fund. Then, \(20000 - s\) is the amount invested in the bond.

The total return is the sum of the returns from each investment. Total Return = (Return from Stocks) + (Return from Bonds) Total Return = \(0.08s + 0.03(20000 - s)\)

The goal is for the total return to be at least €1,000. \(0.08s + 0.03(20000 - s) \geq 1000\)

Now, solve the inequality: \(0.08s + 600 - 0.03s \geq 1000\) \(0.05s + 600 \geq 1000\) \(0.05s \geq 400\) \(s \geq \frac{400}{0.05}\) \(s \geq 8000\)

The minimum amount the investor must put into the stock fund is €8,000.