Solved Exercises on Inequality Word Problems in Pre-algebra

Master inequality word problems: understanding how to translate real-world scenarios into mathematical inequalities and solve them through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Shopping budget problem
Exercise 1
Maria has $50 to spend on books. Each book costs $8. Write an inequality that represents the maximum number of books she can buy. Solve and interpret the solution.
Definition:

Word problem translation: Converting real-world scenarios into mathematical inequalities by identifying the variable, operation, and constraint.

Problem-solving method:
  1. Identify the unknown quantity (variable)
  2. Find the relationship and constraint
  3. Translate keywords into mathematical symbols
  4. Write the inequality
  5. Solve the inequality
  6. Interpret the solution in context
Define variable
Let \(b\) = number of books
Set up inequality
\(8b \leq 50\)
Solve
\(b \leq 6.25\)
Step 1: Identify the unknown

Unknown: Number of books Maria can buy → Assign variable \(b\)

Step 2: Identify the constraint

Constraint: Total cost cannot exceed $50

Step 3: Write the relationship

Cost per book × Number of books ≤ Budget

\(8b \leq 50\)

Step 4: Solve the inequality

Divide both sides by 8: \(\frac{8b}{8} \leq \frac{50}{8}\)

So: \(b \leq 6.25\)

Step 5: Interpret in context

Since Maria can't buy a fraction of a book, she can buy at most 6 books

\(b \leq 6.25\), so Maria can buy at most 6 books
Final answer:

Maria can buy at most 6 books. The inequality is \(8b \leq 50\), and the solution is \(b \leq 6.25\).

Applied rules:

• Keyword translation: "maximum" → \(\leq\), "cannot exceed" → \(\leq\)

• Variable assignment: Choose a meaningful letter for the unknown

• Context interpretation: Consider practical limitations (whole books)

2 Distance and time problem
Exercise 2
A runner wants to finish a race in less than 3 hours. If the race is 24 miles long, what must be the runner's average speed? Write and solve an inequality.
Definition:

Rate-time-distance relationship: Distance = Rate × Time, so Rate = Distance ÷ Time. When time is constrained, rate is inversely related to time.

Formula
Distance = Rate × Time
Constraint
Time < 3 hours
Inequality
Rate > 8 mph
Step 1: Define the variable

Let \(r\) = average speed in miles per hour

Step 2: Set up the relationship

Time = Distance ÷ Rate, so Time = \(\frac{24}{r}\)

Step 3: Apply the constraint

Time must be less than 3 hours: \(\frac{24}{r} < 3\)

Step 4: Solve the inequality

Multiply both sides by \(r\) (positive, so sign stays same): \(24 < 3r\)

Divide both sides by 3: \(8 < r\), or \(r > 8\)

Step 5: Interpret the solution

The runner must average more than 8 miles per hour to finish in under 3 hours

\(r > 8\)
Final answer:

The runner must average more than 8 miles per hour. The inequality is \(\frac{24}{r} < 3\), and the solution is \(r > 8\).

Applied rules:

• Distance-rate-time formula: \(d = rt\) rearranged appropriately

• Sign preservation: Multiplying by positive number preserves inequality direction

• Real-world interpretation: Speed must be greater than calculated minimum

3 Group activity problem
Exercise 3
A school needs to transport at least 120 students. Each bus holds 30 students. What is the minimum number of buses needed? Write and solve an inequality.
Definition:

Minimum requirement: An inequality using \(\geq\) symbol to represent "at least" or "minimum" conditions in real-world scenarios.

Variable
Let \(n\) = number of buses
Inequality
\(30n \geq 120\)
Solution
\(n \geq 4\)
Step 1: Identify the unknown

Unknown: Number of buses needed → Assign variable \(n\)

Step 2: Identify the requirement

Requirement: Transport at least 120 students

Step 3: Set up the relationship

Students transported = Buses × Students per bus

So: \(30n \geq 120\)

Step 4: Solve the inequality

Divide both sides by 30: \(\frac{30n}{30} \geq \frac{120}{30}\)

So: \(n \geq 4\)

Step 5: Interpret the result

The school needs at least 4 buses (could use more)

\(n \geq 4\)
Final answer:

The school needs at least 4 buses. The inequality is \(30n \geq 120\), and the solution is \(n \geq 4\).

Applied rules:

• Keyword translation: "at least" → \(\geq\)

• Whole number interpretation: Buses must be whole numbers

• Minimum requirement: The smallest whole number satisfying the inequality

Rules and methods, laws,...
\(ax + b < c\)
Linear Inequality
At most
\(x \leq a\)
Maximum allowed
At least
\(x \geq a\)
Minimum required
Less than
\(x < a\)
Strict upper bound
More than
\(x > a\)
Strict lower bound
Key definitions:

Word Problem Inequality: A mathematical inequality derived from a real-world scenario that represents constraints, limits, or requirements.

Translation: The process of converting verbal descriptions into mathematical expressions using variables and inequality symbols.

Constraint: A limiting condition in a problem that restricts the possible values of the variable.

Feasible Solution: A solution that satisfies all constraints of the problem.

Practical Interpretation: Adjusting the mathematical solution to fit real-world requirements (like whole numbers).

Complete methodology:
  1. Read the problem carefully: Understand the scenario and what is being asked
  2. Identify the unknown: Determine what quantity is unknown and assign a variable
  3. Find the constraint: Identify the limiting condition or requirement
  4. Translate to math: Convert the constraint into an inequality
  5. Solve the inequality: Use algebraic methods to isolate the variable
  6. Interpret the solution: Consider real-world implications and constraints
Tip 1: Look for keywords like "at most", "at least", "no more than", "no less than"
Tip 2: Always define your variable clearly before setting up the inequality
Tip 3: Check if your solution makes sense in the real-world context
Tip 4: Consider whether the solution needs to be rounded or restricted to whole numbers
Common errors: Misinterpreting keywords, setting up wrong inequality direction, not considering practical constraints, forgetting to verify solution.
Memory aids: "At most" means ≤, "At least" means ≥, "More than" means >, "Less than" means <.
Solution: Exercises 4 to 5
4 Earnings and savings problem
Exercise 4
Emma earns $12 per hour. She wants to earn at least $180 this week. How many hours must she work? Write and solve an inequality.
Definition:

Minimum earnings: A problem requiring an inequality with \(\geq\) to represent earning at least a specified amount.

Variable
Let \(h\) = hours worked
Inequality
\(12h \geq 180\)
Solution
\(h \geq 15\)
Step 1: Define the variable

Let \(h\) = number of hours Emma works

Step 2: Set up the relationship

Total earnings = Hourly rate × Hours worked

Total earnings = \(12h\)

Step 3: Apply the requirement

Earnings must be at least $180: \(12h \geq 180\)

Step 4: Solve the inequality

Divide both sides by 12: \(\frac{12h}{12} \geq \frac{180}{12}\)

So: \(h \geq 15\)

Step 5: Interpret the solution

Emma must work at least 15 hours to earn $180 or more

\(h \geq 15\)
Final answer:

Emma must work at least 15 hours. The inequality is \(12h \geq 180\), and the solution is \(h \geq 15\).

Applied rules:

• Keyword translation: "at least" → \(\geq\)

• Earnings formula: Total earnings = Rate × Time

• Division property: Dividing by positive number preserves inequality

5 Weight capacity problem
Exercise 5
A delivery truck can carry at most 2,000 pounds. If each package weighs 25 pounds, what is the maximum number of packages that can be loaded? Write and solve an inequality.
Definition:

Maximum capacity: A problem using \(\leq\) to represent an upper limit or maximum allowable value.

Variable
Let \(p\) = number of packages
Inequality
\(25p \leq 2000\)
Solution
\(p \leq 80\)
Step 1: Identify the variable

Let \(p\) = number of packages

Step 2: Establish the relationship

Total weight = Packages × Weight per package

Total weight = \(25p\)

Step 3: Apply the constraint

Total weight must be at most 2000 pounds: \(25p \leq 2000\)

Step 4: Solve the inequality

Divide both sides by 25: \(\frac{25p}{25} \leq \frac{2000}{25}\)

So: \(p \leq 80\)

Step 5: Contextualize the solution

The truck can carry at most 80 packages (not more than 80)

\(p \leq 80\)
Final answer:

The truck can carry at most 80 packages. The inequality is \(25p \leq 2000\), and the solution is \(p \leq 80\).

Applied rules:

• Keyword translation: "at most" → \(\leq\)

• Multiplication relationship: Total = Units × Value per unit

• Capacity constraint: Upper limit on total quantity

Comprehensive Summary: Inequality Word Problems
\(ax + b \leq c\)
Constrained Optimization
Key definitions:

Inequality Word Problem: A real-world scenario that can be modeled using a mathematical inequality to represent constraints, limits, or requirements.

Translation: The process of converting verbal descriptions into mathematical expressions and inequalities.

Constraint: A condition that limits the possible values of a variable in a problem.

Feasible Region: The set of all values that satisfy the constraints of the problem.

Optimization: Finding the maximum or minimum value within the feasible region.

Complete methodology:
  1. Problem comprehension: Read the entire problem carefully to understand the scenario
  2. Variable identification: Determine what quantity is unknown and assign a variable
  3. Relationship analysis: Identify how quantities relate to each other
  4. Keyword recognition: Identify words that indicate inequality symbols
  5. Model construction: Build the mathematical inequality
  6. Solution: Solve the inequality algebraically
  7. Interpretation: Relate the solution back to the real-world context
Tip 1: Keywords for ≤: "at most", "no more than", "maximum", "up to"
Tip 2: Keywords for ≥: "at least", "no less than", "minimum", "greater than or equal"
Tip 3: Keywords for <: "less than", "fewer than", "under", "below"
Tip 4: Keywords for >: "more than", "greater than", "over", "above", "exceeds"
Tip 5: Always verify your solution fits the real-world constraints
Common errors: Misreading inequality direction, not defining variables clearly, forgetting to consider practical constraints, misinterpreting keywords.
Memory aids: "At most" means ≤, "At least" means ≥, "No more" means ≤, "No less" means ≥.
Essential rules to remember:

• Keyword identification: Recognize inequality indicators in word problems

• Variable definition: Clearly state what your variable represents

• Model accuracy: Ensure the inequality correctly represents the constraint

• Solution interpretation: Consider real-world applicability of the solution

• Verification: Check that your solution satisfies the original problem

Visualization: Constraint Relationships
Exercise 6: Constraint Visualization
Visualizing different constraint types:
\(f_1(x): 8x \leq 50\) (budget constraint)
\(f_2(x): \frac{24}{x} < 3\) (time constraint)
\(f_3(x): 30x \geq 120\) (minimum requirement)

Analysis: The chart shows how different constraints create different feasible regions.

  • Upper bound constraints limit maximum values
  • Lower bound constraints establish minimum requirements
  • Ratio constraints create inverse relationships
  • Each constraint defines a solution boundary

Questions & Answers

Question: I always get confused about whether to use ≤ or <, or ≥ or >. How do I know which one to use?

Answer: The key is in the wording of the problem:

For ≤ (less than or equal): Look for words like "at most", "no more than", "maximum", "up to", "can spend up to".

  • "Maria can spend at most $50" → Total ≤ 50
  • "The truck can carry no more than 2000 lbs" → Weight ≤ 2000

For ≥ (greater than or equal): Look for words like "at least", "no less than", "minimum", "must be at least".

  • "Emma must earn at least $180" → Earnings ≥ 180
  • "They need no less than 120 students" → Students ≥ 120

For < (strictly less than): Look for "less than", "fewer than", "under", "below".

  • "Finish in less than 3 hours" → Time < 3

For > (strictly greater than): Look for "more than", "greater than", "exceeds", "over".

  • "Average more than 8 mph" → Speed > 8

The words "at most" and "at least" are your biggest clues! "At most" means that value is the maximum allowed (≤), and "at least" means that value is the minimum required (≥).

Question: What if the solution is a decimal, but the problem is about whole things like people or items?

Answer: You must adjust your answer to fit the real-world context:

For maximum constraints (≤): Round down to the nearest whole number.

  • If \(b \leq 6.25\) (books), then Maria can buy at most 6 books (round down)
  • You can't buy 6.25 books, so 6 is the largest whole number that satisfies the constraint

For minimum constraints (≥): Round up to the nearest whole number.

  • If \(n \geq 4.2\) (buses), then you need at least 5 buses (round up)
  • 4 buses wouldn't meet the requirement, so you need 5 to satisfy \(n \geq 4.2\)

Always consider: "Does my answer make sense in the real-world scenario?" If you get 6.25 books, the answer is 6. If you get 4.2 buses, the answer is 5.

The mathematical solution is \(x \leq 6.25\), but the practical solution is "at most 6 books".

Question: How do I check if my answer is correct for a word problem?

Answer: There are two levels of checking:

Level 1: Mathematical check: Substitute your solution back into the original inequality.

  • For \(8b \leq 50\) with solution \(b \leq 6.25\), test \(b = 6\): \(8(6) = 48 \leq 50\) ✓
  • Test the boundary: \(b = 6.25\): \(8(6.25) = 50 \leq 50\) ✓

Level 2: Contextual check: Verify the answer makes sense in the real-world scenario.

  • Can Maria buy 6 books? Yes, and she'll spend $48, which is under $50
  • Could she buy 7 books? No, that would cost $56, exceeding her budget

Always ask: "Does this answer address the question asked?" and "Is this practical in the given scenario?"

For example, if you find that \(h \geq 15\) for hours to work, verify that working 15 hours earns exactly $180, and working more earns more than $180.