Solve System of Inequalities - Word Problem - Algebra-1

1. Fundamental Concepts

System of Inequalities: A set of multiple inequalities containing the same variables. Its solution is the range (or region) of variable values that satisfy all inequalities simultaneously.
Word Problem: A problem that describes quantitative relationships through real-life scenarios. It needs to be converted into a system of inequalities for solving, with results interpreted in the context of the scenario.
Definition of Variables: Unknown quantities (such as "quantity," "cost," "time") are represented by letters (e.g., ).
Key Words for Inequality Relationships: Identify words indicating inequality in the scenario, for example:
- "At least" / "No less than" →
- "At most" / "No more than" →
- "More than" →
- "Less than" →
- "Non-negative" → (e.g., quantity, time cannot be negative)

2. Key Concepts

Conversion Steps: The core logic for converting verbal descriptions into a system of inequalities:
1. Determine variables (clarify the actual meaning of each variable);
2. Identify all constraints (e.g., resource limits, quantitative relationships, non-negativity);
3. Express each constraint with an inequality to form a system of inequalities.
Practical Significance of Solutions: The solution to the system of inequalities must conform to real-life scenarios (e.g., the number of people is a positive integer, time is non-negative). Mathematical solutions may need to be filtered.
Solutions to Multivariable Inequality Systems: For two variables (e.g., ), the solution set is a region in the coordinate plane; for one variable, it is an interval on the number line.

3. Examples

Easy

Problem: Xiaoming wants to buy notebooks and pens. Each notebook costs 5 yuan, and each pen costs 8 yuan. He has 50 yuan, and he wants to buy at least 3 notebooks. Let x be the number of notebooks and y be the number of pens ( are non-negative integers).

Inequality System:

Problem: A student wants to buy candies and chocolates. Candies cost 2 dollars each, and chocolates cost 3 dollars each. He has 20 dollars and wants to buy more candies than chocolates. Let x = number of candies, y = number of chocolates (\(x, y\) are non-negative integers).

Inequality System:

Medium

Problem: A factory produces two types of parts, A and B. Part A yields a profit of 30 yuan per unit and takes 2 hours to produce; Part B yields 20 yuan per unit and takes 1 hour to produce. The total daily working hours cannot exceed 10 hours, and at least 2 units of Part A must be produced daily. Let x be the number of Part A and y be the number of Part B produced daily ( are non-negative integers).

Inequality System:

Hard

Problem: A travel agency organizes tourists to visit places A and B. The ticket for place A is 50 yuan per person, and for place B is 80 yuan per person. The agency can receive at most 50 people, and the number of people visiting A must be at least 1.5 times that of B. Let x be the number of people visiting A and y be the number visiting B ( are positive integers). The total ticket revenue must be at least 3000 yuan.

Inequality System:

4. Problem-Solving Techniques

Clarify Variables: Represent unknowns with concise letters, specifying units and value ranges (e.g., "x is a positive integer").
Extract Constraints: Analyze the problem sentence by sentence to identify all constraints (e.g., resources, quantities, relationships), avoiding omissions (especially non-negativity).
Convert to Inequalities: Accurately translate keywords into inequalities (e.g., "no more than" corresponds to ), paying attention to expressions of "sum," "difference," and "multiple."
Solve the System:
- For single-variable systems: Solve by transposing and combining like terms, and represent the solution set on a number line;
- For two-variable systems: Plot the region of each inequality on a coordinate system; the intersection is the solution set.
Filter by Reality: Exclude solutions that do not fit the scenario (e.g., negative numbers, non-integers). If necessary, list all possible solutions for verification.
Verify Results: Substitute solutions back into the original problem to check if all conditions are satisfied, ensuring logical consistency.