6.6 Creating Equations, Inequalities, & Constraint Models

Creating Equations, Inequalities, & Constraint Models

Solving an equation is mechanical. Writing the equation is where the points are, and Level A tests it as a distinct skill across three standards: create equations and inequalities in one variable; create equations in two or more variables and graph them "on coordinate axes with labels and scales"; and represent constraints by equations, inequalities, or systems, "interpreting solutions as viable or non-viable options in a modeling context."

Part 1: One-Variable Equations and Inequalities

The reliable procedure is four steps, and skipping step 1 is what causes most errors.

1. Define the variable in words. "Let $h$ = the number of hours of labor." Not just "let x = hours" — write the unit.
2. Translate one sentence at a time.
3. Solve.
4. Answer the question that was asked, which is often not the variable you solved for.

Problem. A repair shop charges a $65 diagnostic fee plus $95 per hour. A customer's total bill was $445. How long did the repair take?

$$95h = 380 \;\Rightarrow\; h = \mathbf{4 \text{ hours}}$$

The same problem as an inequality

The customer can spend at most $445. What is the greatest number of hours she can afford?

$$95h + 65 \le 445 \;\Rightarrow\; 95h \le 380 \;\Rightarrow\; h \le 4$$

Translate the boundary language carefully — this is a scored distinction:

Part 2: Two-Variable Equations and Their Graphs

When two quantities vary together, the model is a two-variable equation and the answer is a graph, not a number.

A landscaping crew charges $40 per lawn and $25 per hedge. Write an equation for the revenue $R$ from $\ell$ lawns and $h$ hedges.

$$R = 40\ell + 25h$$

If total revenue is fixed at $400, the equation $40\ell + 25h = 400$ describes every combination that produces that revenue, and its graph is a line segment.

Graphing with labels and scales

The standard explicitly requires labels and scales, and TABE items ask about them:

• Label each axis with the quantity and its unit — "Lawns mowed," "Revenue (dollars)" — not just "x" and "y."
• Choose a scale that fits the data. If revenue runs to $400, tick every $50, not every $1.
• Start at a sensible origin. For counts of jobs, both axes begin at 0.
• Plot the intercepts first. Setting $h = 0$ gives $\ell = 10$; setting $\ell = 0$ gives $h = 16$. Connect $(10, 0)$ and $(0, 16)$.

Part 3: Constraints and Systems

A constraint is a limit the situation imposes. Each one becomes an inequality, and together they carve out a region of acceptable solutions.

Problem. A food-truck owner makes tacos and burritos. Each taco uses 2 oz of beef and each burrito uses 5 oz. She has 200 oz of beef. She must make at least 20 tacos, and she cannot make a negative number of anything.

The set of $(t, b)$ pairs satisfying all three at once is the feasible region. Any point inside it is a workable production plan; any point outside violates something.

Testing a candidate plan. Is 30 tacos and 30 burritos feasible?

• $2(30) + 5(30) = 60 + 150 = 210$, and $210 \le 200$ is false.

The plan fails on beef supply, so it is rejected — even though $t \ge 20$ and $b \ge 0$ both hold. All constraints must be satisfied simultaneously.

Part 4: Viable vs. Non-Viable Solutions

This is the modeling judgment TABE actually asks about, and it is the difference between an algebra answer and a correct answer.

Worked viability check. A ticket equation gives $a = 8.4$ adult tickets and $c = 11.6$ child tickets. Tickets are whole objects, so this exact solution is non-viable. The correct response is to report that no whole-number solution exists for the stated totals, or to identify the nearest whole-number pair that satisfies the constraints — not to round silently and present 8 and 12 as if they solved the original system.

The habit to build: after every modeling answer, ask two questions. Can this quantity be negative? Can it be a fraction? If the answer to either is no and your solution says otherwise, the model or the arithmetic needs another look.