Inequality Word Problems - Math 6

1. Fundamental Concepts

Definition: Inequality word problems involve translating real-world situations into mathematical inequalities to solve for unknowns.
Types of Inequalities: Less than (<), less than or equal to (≤), greater than (>), and greater than or equal to (≥).
Solution Set: The set of all possible values that satisfy the inequality.

2. Key Concepts

Keyword Correspondence: Specific phrases in problems indicate inequality relationships, and "or equal to" must be clearly stated:
- Greater than: more than, above, past, exceed
- Less than: less than, below, under, fall short of
- Greater than or equal to: at least, no less than, a minimum of
- Less than or equal to: up to, no more than, a maximum of
Core Distinction: Use an equation when the problem describes a single definite value, and use an inequality when it describes a range of values.

3. Examples

Example 1 (Basic)

Problem 1: A roller coaster at an amusement park has a rule: riders must be taller than 1.2 meters to ride alone. Let h represent a rider’s height (in meters). Write an inequality to describe the height requirement for riding alone, and explain why the endpoint is excluded.

Solution:

Explanation:

Identify the key constraint: The phrase "taller than 1.2 meters" means a rider’s height must exceed 1.2 meters—exactly 1.2 meters does not meet the requirement.
Match the phrase to the inequality sign: "Taller than" (strictly greater than) corresponds to , which does not include the endpoint (1.2 meters).

Problem 2: A bookstore requires customers to spend at least 15 dollars to get a free bookmark. Let m represent the amount a customer spends (in dollars). Write an inequality to describe this rule.

Solution:

Explanation:

Identify the key phrase: "at least 15 dollars" means the spending amount cannot be less than 15, and 15 itself is a qualifying value.
Match the phrase to the inequality sign: "at least" corresponds to .
Define the variable: m stands for the spending amount.
Construct the inequality: Combine the variable and sign with the threshold value (15) to get .

Example 2 (Intermediate)

Problem: A delivery truck can carry a maximum load of 800 kilograms. Each box of goods weighs 25 kilograms, and the truck itself has a fixed weight of 150 kilograms. Let n be the number of boxes loaded. Write an inequality to find the maximum number of boxes the truck can carry.

Solution:

Explanation:

Analyze the total weight composition: The total weight the truck bears includes the fixed weight of the truck (150 kg) and the weight of the boxes.
Calculate the weight of the boxes: Each box is 25 kg, so n boxes weigh 25n kg.
Identify the key constraint: "Maximum load of 800 kilograms" means the total weight cannot exceed 800 kg, so use .
Construct the inequality: Add the fixed weight and box weight, then set the sum less than or equal to the maximum load: .

4. Problem-Solving Techniques

Analyze the Problem: Determine whether the problem involves a range of values (for inequalities) or a single fixed value (for equations).

Extract Key Phrases: Identify words like "at least" or "less than" to confirm the direction of the inequality sign.

Define Variables: Assign a letter to represent the unknown quantity (e.g., number of items, time, distance).

Construct the Inequality: Translate the practical relationship described in the problem into a mathematical inequality, including constants and coefficients related to the variable.

Graph the Solution: Plot the solution set on a number line—use an open circle for or (excluding the endpoint) and a closed circle for or (including the endpoint), then shade the relevant region.