Domain: Algebra | Skill: Linear inequalities in one or two variables | Difficulty: Medium

Conquering Constraints: Your Guide to Medium-Level SAT Linear Inequalities

Welcome to the MyTestPrep.ai guide to SAT Math! Today, we’re tackling a common question type from the Algebra domain: Linear Inequalities in one or two variables. These questions test your ability to turn real-world scenarios—like budgets, shipping limits, or business plans—into mathematical expressions. On the SAT, they are a fantastic way to test your logical reasoning alongside your algebra skills. While they might look intimidating, mastering the strategies for Medium-difficulty inequality questions is a reliable way to secure your success on the math section.

Deconstructing the Questions: Common Formats

Linear inequality questions can appear in a few different formats. Understanding these patterns is the first step to solving them quickly and accurately. Here’s a breakdown of what you can expect:

Typical Format	What It Tests	Quick Strategy
A company offers two plans… A farmer is planting two crops…	Setting up a system of inequalities from a word problem with multiple constraints.	Define your variables first (e.g., \(x\) for Plan A, \(y\) for Plan B). Translate each sentence into a separate inequality.
The point \((x, y)\) is a solution to the system… Which of the following is also a solution?	Understanding what it means for a point to be a solution (i.e., it satisfies all inequalities).	Plug the coordinates of each answer choice into the inequalities. The one that makes all statements true is the answer.
If [condition], what is the maximum/minimum possible value of [variable]?	Optimizing one variable within the constraints of the system.	To maximize one variable, you often need to minimize the other variables it’s related to (and vice versa).

Real SAT-Style Example

Let’s look at a typical Medium-difficulty problem. This question combines multiple constraints and asks for a maximum value, making it a perfect example of what to expect.

A shipping container can hold packages and crates. Each package weighs \(50\) pounds, and each crate weighs \(150\) pounds. The container can hold at most \(2,000\) pounds. The number of packages cannot exceed \(30\), and the number of crates cannot exceed \(10\). If at least \(20\) packages must be included, what is the maximum number of crates that can be loaded into the container?

A) 5

B) 6 ✅

C) 10

D) 13

Step-by-Step Solution:

1. Define Variables: Let \(p\) be the number of packages and \(c\) be the number of crates.
2. Set Up Inequalities:

• Weight constraint: \(50p + 150c \le 2000\)
• Package constraint 1: \(p \le 30\)
• Package constraint 2: \(p \ge 20\)
• Crate constraint: \(c \le 10\)

3. Analyze the Goal: We want to find the maximum number of crates (\(c\)). To maximize \(c\), we need to use the least amount of weight possible for the packages.
4. Use the Limiting Constraint: The problem states there must be at least 20 packages (\(p \ge 20\)). The minimum number of packages we can have is 20. Let’s use this value for \(p\) to see the maximum number of crates we can fit.
5. Solve for \(c\): Substitute \(p = 20\) into the weight inequality: \[50(20) + 150c \le 2000\] \[1000 + 150c \le 2000\] \[150c \le 1000\] \[c \le \dfrac{1000}{150}\] \[c \le 6.66…\]
6. Final Answer: Since the number of crates must be a whole number, the maximum value for \(c\) is 6. This also satisfies the condition that \(c \le 10\). Therefore, 6 is the correct answer.

Your 4-Step Strategy for Linear Inequalities

Follow this methodical approach to confidently solve any Medium-level inequality problem.

1. Identify All Variables and Constraints: Before writing any equations, carefully read the problem and define your variables (e.g., \(x\), \(y\)). Then, list out every rule, limit, or condition in plain English.
2. Translate Words into Math: Convert each constraint into a precise mathematical inequality. Pay close attention to keywords: “at most” means \(\le\), “at least” means \(\ge\), “cannot exceed” means \(\le\), and “more than” means \(>\).
3. Isolate and Solve for the Target Variable: Determine what the question is asking for—a maximum, a minimum, or a possible value. Use substitution from the other constraints to isolate the target variable and solve.
4. Check Your Answer Against All Constraints: Once you have a solution, quickly plug it back into the scenario to ensure it doesn’t violate any of the original rules. This final check prevents simple mistakes.

Applying the Strategy to Our Example

Let’s use our 4-step strategy on the shipping container problem to see how it works in practice.

Step 1 Applied: Identify Variables and Constraints

First, we define our variables: \(p = \text{number of packages}\) and \(c = \text{number of crates}\). Then, we identify the constraints from the text:

• Total weight is at most 2,000 pounds.
• Number of packages is 30 or fewer.
• Number of crates is 10 or fewer.
• Number of packages is at least 20.
• Goal: Find the maximum possible value for \(c\).

Step 2 Applied: Translate into Math

Now, we convert our list of constraints into a system of inequalities:

\[50p + 150c \le 2000\] \[p \le 30\] \[c \le 10\] \[p \ge 20\]

Step 3 Applied: Isolate and Solve

Our goal is to maximize \(c\). The weight inequality, \(50p + 150c \le 2000\), shows that as \(p\) increases, the maximum possible \(c\) decreases. To maximize \(c\), we need to choose the smallest possible value for \(p\). The constraint \(p \ge 20\) tells us the minimum value for \(p\) is 20. We substitute \(p=20\) into the main inequality:

\[50(20) + 150c \le 2000\] \[1000 + 150c \le 2000\] \[150c \le 1000\] \[c \le 6.66…\]

Since \(c\) must be a whole number, the maximum integer value is 6.

Step 4 Applied: Check Your Answer

Does the solution \(c=6\) (with \(p=20\)) work with all constraints?

• Weight: \(50(20) + 150(6) = 1000 + 900 = 1900\). Is \(1900 \le 2000\)? Yes.
• Packages: Is \(20 \le 30\)? Yes. Is \(20 \ge 20\)? Yes.
• Crates: Is \(6 \le 10\)? Yes.

All conditions are satisfied. Our answer is correct.

Ready to Try It on Real Questions?

Now that you understand the strategy, it’s time to practice with authentic SAT questions! Head to https://mytestprep.ai(mytestprep.ai) and follow these steps:

1. Login using your account or signup on mytestprep.ai
2. Click on Practice Sessions once you are on the dashboard. You will see the link on the left side navigation menu of the dashboard
3. Click on Create New Session
4. Start with Co-Pilot Mode on with hints and explanations—it’s like having a personal coach who explains exactly why each answer is right or wrong
5. Once comfortable, switch to Timed Mode to build speed
6. Start practicing. Happy Practicing!

Key Takeaways

• Translate Keywords Carefully: “At most” (\(\le\)), “at least” (\(\ge\)), “no more than” (\(\le\)), and “fewer than” (\(<\)) have precise mathematical meanings. Don’t mix them up!
• Maximize by Minimizing: To find the maximum value for one variable in a shared constraint, you often need to use the minimum possible value of the other variable(s).
• Define Variables First: Always start by assigning letters (like \(p\) and \(c\)) to the quantities in the problem. This simple step prevents confusion.

Always Double-Check: A quick check at the end ensures your answer is logical and satisfies all conditions of the problem, not just the one you used to solve.