SAT-Math-15: Modeling Real-World Scenarios with Inequalities
This lesson teaches students how to construct linear inequalities from real-world constraints, budgets, and minimum/maximum requirements expressed in word problems.
1. Translating Constraint Keywords
In word problems, real-world constraints translate into inequality symbols rather than equal signs. The SAT loves testing your ability to spot the difference between a maximum ceiling and a minimum floor.
Here is your translation dictionary for constraints:
"At most", "maximum of", "cannot exceed", "up to" $\rightarrow \le$
"At least", "minimum of", "no less than" $\rightarrow \ge$
"More than", "exceeds" $\rightarrow >$
"Less than", "under" $\rightarrow <$
⚠️ The Inverse Intuition Trap: > When students hear the word "maximum," their brain automatically thinks of the "greater than" ($>$) concept. But think about it practically: if the maximum number of people allowed in a room is 50, the number of people must be less than or equal to 50 ($x \le 50$).
🇺🇿 Uzbek Explanation:
Cheklovchi kalit so'zlar: Matnli masalalarda eng ko'p adashtiradigan so'zlar bu "at most" va "at least" iboralaridir:
At most (ko'pi bilan): Bu belgi kichik yoki teng ($\le$) deganidir. Masalan, "ko'pi bilan 100 dollar sarqlay olaman" degani xarajatlar $\le 100$ bo'lishi kerak degani.
At least (kamida): Bu belgi katta yoki teng ($\ge$) deganidir. Masalan, "imtihondan o'tish uchun kamida 70 ball olishingiz kerak" ballar $\ge 70$ bo'lishini anglatadi.
2. Setting Up the Inequality Model
Real-world inequality models typically follow a structured format:
$$\text{Rate}_1 \cdot x + \text{Rate}_2 \cdot y \quad [\text{Symbol}] \quad \text{Total Constraint}$$
Let's see how this structure applies to a high-yield scenario.
3. High-Yield SAT Practice Question
Question: A catering company prepares box lunches for a corporate seminar. Each chicken lunch costs $8 to prepare, and each vegetarian lunch costs $6 to prepare. The caterer has a total budget constraint: they can spend a maximum of $450 on lunch preparation today. The caterer must prepare at least 40 chicken lunches.Which of the following systems of inequalities models this situation, where $c$ represents the number of chicken lunches and $v$ represents the number of vegetarian lunches?A) $\begin{cases} 8c + 6v \le 450 \\ c \ge 40 \end{cases}$B) $\begin{cases} 8c + 6v \ge 450 \\ c \le 40 \end{cases}$C) $\begin{cases} 8c + 6v < 450 \\ c > 40 \end{cases}$D) $\begin{cases} 6c + 8v \le 450 \\ c \ge 40 \end{cases}$Step-by-Step Explanation:Model the budget constraint:Each chicken lunch ($c$) costs $8 $\rightarrow 8c$
Each vegetarian lunch ($v$) costs $6 $\rightarrow 6v$The total combined cost is $8c + 6v$.The keyword is "maximum of $450", which translates to $\le 450$.
First inequality: $8c + 6v \le 450$. (This eliminates Option B and Option C).
Model the quantity constraint:
The caterer must prepare "at least 40" chicken lunches ($c$).
"At least" translates to $\ge 40$.
Second inequality: $c \ge 40$.
Match with the choices: Option A matches our constructed system perfectly. Option D accidentally flips the costs of the items.
Correct Answer: A
Summary
Translate "at most" to $\le$ and "at least" to $\ge$.
Map cost, weight, or time rates directly to their respective variables before combining them against the total constraint.
Read carefully to ensure rates are matched with the correct variables.
You're completely mastering this block, my bro! Ready to advance to SAT-Math-16: Systems of Linear Equations: Solving by Substitution?