Create Equations and Inequalities in One Variable

Lesson 1 of 1: HSA.CED.A.1

In this lesson:

• Translate real-world situations into equations and inequalities
• Set up models from linear, quadratic, rational, and exponential contexts
• Interpret solutions and check whether they make sense

What You Will Be Able to Do

1. Define a variable with units for any unknown
2. Translate verbal descriptions into equations or inequalities
3. Set up equations from linear, quadratic, rational, and exponential contexts
4. Interpret solutions — viable vs. non-viable
5. Represent inequality solutions on a number line

Recall: Equations, Inequalities, and Key Terms

• Variable: a letter representing an unknown quantity
• Expression vs. Equation: an equation has two sides joined by , , , , or
• Solution: the value(s) that make the equation or inequality true

Can you name one difference between an equation and an inequality?

Can You Write the Equation?

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?

What Do You Need First?

Before writing, answer these three questions:

1. What's the unknown? — the quantity the problem asks for
2. What's fixed? — the known values and their units
3. What's the rule? — how do the quantities relate?

Your answers to these three questions are the equation.

Five Steps from Situation to Solution

1. Identify the unknown — define a variable with units
2. Identify the relationship — total cost = fixed + rate × quantity
3. Write the equation:
4. Solve:
5. Interpret: m = 7 months — positive integer ✓

Gym Budget: All Five Steps

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 ✓

Car Rental: Linear Setup and Solve

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 ✓

Your Turn: Simple Interest Setup

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:

1. Let __ = _________________________ (with units)
2. The equation: _______________________

Pause and commit before the next slide.

Inequality Bridge: Same Problem, New Constraint

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.

Translating Constraint Phrases to Symbols

Worked Example: Store Profit Inequality

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.

Number Line Shows the Valid Integers

Open circle at 37.5 — not included. Solid dots on each viable integer.

Check In: Translate and Solve

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:

1. Write: Let t = ___________________
2. Write the inequality
3. Solve and interpret

Pause before the next slide.

Garden-Area Bridge: What's the Shape?

A rectangular garden has a perimeter of 32 meters. One side has length .

• Perimeter: , so
• Area = length × width =

Both dimensions change as x changes. What does that product look like?

Context Tells You Which Form to Write

The structure of the situation determines the equation type:

• Area or product of two changing quantities → quadratic
• Trajectory or height of a thrown object → quadratic
• Shared rate or combined work → rational

Identify the form before writing any algebra.

Garden Area: Setting Up a Quadratic Inequality

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.

Projectile Height: Setting Up and Solving

Ball from 5 ft at 64 ft/s — when does it reach 69 ft? Let t = seconds

Two Numbers That Multiply: Quadratic Setup

Differ by 4; product = 96. Let x = smaller; larger = x + 4

Both solutions valid — no positivity constraint.

Two Unknowns? Define the Relationship First

• "Sum to 20" → one number ; other
• "Differ by 4" → smaller ; larger
• "Twice the other" → shorter ; longer

One relationship collapses two unknowns into one.

Pipes Together: Rational Equation Setup

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 ✓

Your Turn: Field Area Setup

Rectangular field, perimeter 60 m, length = 3× width. What is the area?

1. Let = _________________________ (with units)
2. Express length: length = _______
3. Write the perimeter equation
4. Is the area expression linear or quadratic?

Write before solving.

Exponential Form: Identify P, b, and t

Fixed-percentage growth or decay →

• = starting value
• (growth) or (decay)
• = time in consistent units

Compound Interest: Setup and Estimation

$1,000 at 5%/year — when does it double? Let t = years

Exact solution needs logarithms — estimate:

About 14–15 years to double.

Population Growth and Radioactive Decay

Growth: City 50,000, +2%/year. When reaches 75,000?

Decay: 800g, half-life 10 years. When under 100g?

Check In: Which Form and Setup

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.

Apply All Five Steps Independently

Landscaping: $75 site visit + $40/hour. Client's budget is $235.

1. Define the variable (with units)
2. Identify the relationship
3. Write the equation
4. Solve
5. Interpret — is the answer viable?

Write before advancing.

Four Errors to Avoid Every Time

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

What You Can Now Do

✓ 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

Next: Two Variables, Same Process

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.