In this lesson:
Can you name one difference between an equation and an inequality?
A gym charges a $40 enrollment fee and $25 per month. Maria has a budget of $215 total.
Question: How many months can she afford?
Before you write anything — what would you need to decide first?
Before writing, answer these three questions:
Your answers to these three questions are the equation.
Problem: Enrollment $40, $25/month, budget $215. How many months?
Let m = number of months Maria attends the gym
Interpret: Maria can afford 7 months. Positive integer — makes sense ✓
Problem: $30/day plus $0.15/mile, bill was $97.50. How many miles?
Let m = number of miles driven
Interpret: 450 miles — positive value, reasonable distance ✓
A savings account starts at $1,200 and earns 3% annual simple interest. The account now has $1,560.
Write just these two things — don't solve yet:
Pause and commit before the next slide.
What if Maria can't go over $215 — but doesn't need to spend exactly $215?
The constraint changes from "equals $215" to "at most $215":
The inequality is just the equation with a rephrased constraint.
Problem: Each unit sells for $12. Fixed costs are $450. When does the store make a profit?
Let u = number of units sold
Interpret: Since units must be whole numbers, sell at least 38 units.
Open circle at 37.5 — not included. Solid dots on each viable integer.
Problem: The speed limit is 65 mph. You drive at 60 mph. How long until you've driven at most 150 miles?
Circle the keyword, then:
Pause before the next slide.
A rectangular garden has a perimeter of 32 meters. One side has length .
Both dimensions change as x changes. What does that product look like?
The structure of the situation determines the equation type:
Identify the form before writing any algebra.
Perimeter = 32 m, one side is x — area must be ≥ 48 m²
Let x = length of one side in meters
Solving this requires factoring — the setup is today's objective.
Ball from 5 ft at 64 ft/s — when does it reach 69 ft? Let t = seconds
Differ by 4; product = 96. Let x = smaller; larger = x + 4
Both solutions valid — no positivity constraint.
One relationship collapses two unknowns into one.
Pipe A: 3 hrs alone; Pipe B: 5 hrs alone. Together?
Let t = hours to fill the tank together
Interpret: Faster than either pipe alone ✓
Rectangular field, perimeter 60 m, length = 3× width. What is the area?
Write before solving.
Fixed-percentage growth or decay →
$1,000 at 5%/year — when does it double? Let t = years
Exact solution needs logarithms — estimate:
About 14–15 years to double.
Growth: City 50,000, +2%/year. When reaches 75,000?
Decay: 800g, half-life 10 years. When under 100g?
For each: name the equation type, define the variable, write the setup.
A. Phone plan: $25/month + $0.10/text, bill was $43. How many texts?
B. Town of 8,000 grows 3%/year. When does it reach 12,000?
Don't solve — just set up.
Landscaping: $75 site visit + $40/hour. Client's budget is $235.
Write before advancing.
Expression ≠ equation — needs an or sign
Wrong direction — circle "at most" first, then write
Skip viability — → 38 whole units
Vague variable — "x = months" → define what x counts
✓ Apply the five-step process to any equation type
✓ Recognize the form: constant rate → linear; product → quadratic; shared rate → rational; fixed % → exponential
✓ Viability check is Step 5 — never skip it
"at most" → ≤; two unknowns → find a relationship first
CED.A.2: Same five-step process — but with two quantities changing at once.
You'll write equations in two variables and graph solutions as lines in the plane.
The setup skills you built today carry forward unchanged.
Click to begin the narrated lesson
Create equations and inequalities