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

Conquering the SAT with Confidence: Linear Inequalities Made Easy

Welcome to your go-to guide for mastering one of the most fundamental concepts on the SAT Math section: Linear Inequalities. These questions test your ability to work with relationships where one value isn’t necessarily equal to another, but is instead greater than, less than, or somewhere in between. Think budgets, height restrictions, or time limits. For the SAT, understanding how to translate these real-world scenarios into mathematical statements is crucial. This article focuses on the Easy difficulty level, helping you build a rock-solid foundation for success.

Decoding the Questions: Common Formats for Linear Inequalities

On the SAT, these questions can appear in a few predictable ways. Getting familiar with the patterns is the first step to solving them quickly and accurately.

Typical Format	What It Tests	Quick Strategy
A word problem describing a scenario with constraints (e.g., “A company can spend at most \$500…”)	Translating words into a mathematical inequality.	Identify keywords like “at least” (\(\ge\)), “no more than” (\(\le\)), “greater than” (\(>\)).
“Which of the following points is a solution to the inequality \(y < 2x + 1\)?”	Testing if an ordered pair \((x, y)\) satisfies a two-variable inequality.	Plug the x and y values from each answer choice into the inequality and see which one makes the statement true.
“If \(5x – 3 > 12\), what is a possible value of \(x\)?”	Solving a linear inequality in one variable.	Isolate the variable just like in an equation, but remember to flip the inequality sign if you multiply or divide by a negative number.

Real SAT-Style Example

Let’s look at a typical problem you might encounter. This question is a classic example of translating a real-world situation into a mathematical constraint.

Question: An amusement park ride allows riders who are taller than \(48\) inches and shorter than \(77\) inches. If Sarah is allowed on the ride, which of the following could be her height?

A) \(48\ \mathrm{inches}\)

B) \(76\ \mathrm{inches}\) ✅

C) \(77\ \mathrm{inches}\)

D) \(78\ \mathrm{inches}\)

---

Solution Walkthrough:

Let \(h\) be Sarah’s height in inches.

The phrase “taller than \(48\) inches” translates to \(h > 48\). It does not include \(48\).

The phrase “shorter than \(77\) inches” translates to \(h < 77\). It does not include \(77\).

We need a value for \(h\) that satisfies both conditions, which we can write as a compound inequality: \(48 < h < 77\).

Now we test the options:

• A) \(48\) is not greater than \(48\).
• B) \(76\) is greater than \(48\) AND less than \(77\). This works.
• C) \(77\) is not less than \(77\).
• D) \(78\) is not less than \(77\).

Therefore, the only possible height for Sarah is \(76\) inches.

Your 4-Step Strategy for Easy Linear Inequalities

Follow these steps to consistently solve these problems with ease and accuracy.

1. Identify the Keywords: Read the problem carefully and circle or underline words that indicate an inequality. Pay close attention to phrases like “at least,” “at most,” “more than,” “less than,” “no more than,” and “no less than.”
2. Translate into a Mathematical Statement: Convert the English phrases into a mathematical inequality. Assign a variable (like \(x\) or \(h\)) to the unknown quantity. Remember: “at least” means \(\ge\) and “at most” means \(\le\).
3. Solve or Test the Choices: If it’s a simple inequality, solve for the variable. If you’re given answer choices, it’s often faster to test each choice against the inequality you created. This is especially true for word problems like our example.
4. Check for Reasonableness: After finding an answer, quickly glance back at the original problem. Does your answer make sense in the context of the story? This final check can help you catch small mistakes, like misinterpreting “taller than” as including the number itself.

Applying the Strategy to Our Example

Let’s break down the amusement park problem using our 4-step strategy. Seeing the steps in action is the best way to make them stick.

Step 1 Applied: Identify the Keywords

First, we read the prompt and pull out the key constraints. The important phrases are “taller than \(48\) inches” and “shorter than \(77\) inches.” These words tell us we are dealing with a range, not exact values.

Step 2 Applied: Translate into a Mathematical Statement

Let’s use \(h\) for Sarah’s height.

• “Taller than \(48\)” means the height must be greater than 48. The symbol for this is \(>\). So, we write \(h > 48\).
• “Shorter than \(77\)” means the height must be less than 77. The symbol for this is \(<\). So, we write \(h < 77\).

Combining these gives us the compound inequality: \(48 < h < 77\).

Step 3 Applied: Solve or Test the Choices

Since this is a multiple-choice question, testing the choices is the most direct path. We check each option against our inequality \(48 < h < 77\):

• Is \(48 < 48 < 77\)? No, \(48\) is not greater than \(48\).
• Is \(48 < 76 < 77\)? Yes, this is true.
• Is \(48 < 77 < 77\)? No, \(77\) is not less than \(77\).
• Is \(48 < 78 < 77\)? No, \(78\) is not less than \(77\).

Option B is the only one that makes the statement true.

Step 4 Applied: Check for Reasonableness

Does a height of 76 inches make sense? Yes. Someone who is 76 inches tall is allowed on a ride that requires you to be taller than 48 inches and shorter than 77 inches. The answer fits the story perfectly.

Ready to Try It on Real Questions?

The best way to build confidence is to apply these strategies to a variety of questions. With mytestprep.ai, you can drill down on this specific skill until it’s effortless.

1. Log in to your mytestprep.ai account.
2. From the dashboard, navigate to the Math section.
3. Choose Algebra → Linear inequalities in one or two variables → Easy.

Start with Tutor Mode to get instant help from our Co-Pilot AI if you’re stuck. It will provide hints and explanations without giving away the answer. When you feel ready, switch to Timed Mode to simulate real test conditions.

Key Takeaways for Linear Inequalities

Remember these core principles as you practice:

• Keywords are traffic signs: Words like “at most” or “greater than” tell you exactly which inequality symbol to use. Don’t rush past them.
• Translate words to math first: Before you try to solve, turn the English sentence into a clean mathematical inequality. This prevents confusion.
• Testing choices is a valid shortcut: For multiple-choice questions, plugging in the answers is often the fastest and safest way to find the solution.
• The Golden Rule: When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is one of the most common traps!