Create linear, quadratic, and exponential equations and inequalities in one or two variables to model and solve problems (NC.M1.A-CED.1, A-CED.2).

What this topic is asking

NC.M1.A-CED.1 asks you to create equations and inequalities in one variable (linear, quadratic, or exponential) to model and solve problems. NC.M1.A-CED.2 asks you to create equations in two variables to represent relationships and graph them. This is the reverse of solving: you build the model from words, then use it.

Defining the variable

Every modeling answer starts by naming the unknown clearly.

A vague variable ("let $x$ be the taxi") causes wrong setups; a precise one ("let $m$ be the number of miles") makes the translation obvious.

Translating the language

The structure of a linear model is almost always rate times quantity plus fixed amount.

Choosing the inequality symbol

Comparison words map to symbols, and getting this mapping right is most of the inequality items.

• "at least," "minimum," "no less than" become $\ge$.
• "at most," "maximum," "no more than" become $\le$.
• "more than," "greater than" become $>$.
• "fewer than," "less than" become $<$.

For example, "you can spend no more than \50$" with items costing$\$8$ each becomes $8n \le 50$.

Judging viability

A modeling answer is not finished until you check it makes sense. A-CED.3 (constraints) and A-CED.1 both expect you to reject non-viable solutions: you cannot buy $-2$ tickets or wait $3.7$ buses if buses come whole. Always ask whether the solution can exist in the real context.

How the NC Math 1 EOC examines this topic

• Multiple choice. Choose the equation or inequality that models a situation.
• Gridded response. Build the model, solve, and enter the value.
• Technology-enhanced. Match models to contexts, or select all true statements about a model.

Creating equations is the inverse of interpreting expressions: there you read meaning from a model, here you write a model from meaning. Both rely on seeing the rate as a coefficient and the fixed amount as a constant. Once built, the model is solved with the equation and inequality skills.

Why modeling is the point of algebra

Algebra earns its place because it turns messy, wordy situations into compact symbols you can manipulate with reliable rules. The model $C = 20t + 30$ answers not one question but every question about that gym's cost: the total after any number of months, the months affordable on a budget, the break-even against another plan. Building the model well, defining the variable, placing the rate and fixed amount, choosing the right relation, is therefore the highest-leverage skill on the test, because it unlocks the rest of the Algebra and Functions categories. The EOC rewards a clear setup even when a calculator does the arithmetic.

Try this

Q1. A phone plan costs \15$plus$\$0.05$ per text. Write an equation for the cost $C$ of $t$ texts. [1 point]

• Cue. $C = 0.05t + 15$ (rate is the coefficient, fee is the constant).

Q2. A club must raise at least \400$and has$\$250$. Write an inequality for the amount $a$ still needed. [2 points]

• Cue. $250 + a \ge 400 \Rightarrow a \ge 150$.