Solved Exercises on Inequalities Word Problems in Integrated Math 1

Master inequalities word problems: real-world applications through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Budget constraint
Exercise 1
Sarah has $50 to spend at the bookstore. She wants to buy books that cost $8 each. Write an inequality to represent the maximum number of books she can buy and solve it.
Definition:

Word problem: A real-world situation expressed in words that requires mathematical modeling

Constraint: A limit or restriction that affects the solution

Variable: The unknown quantity to be determined

Word problem method:

To solve inequalities word problems:

  1. Identify the variable: Determine what quantity needs to be found
  2. Find key information: Identify constraints and relationships
  3. Translate to inequality: Convert words to mathematical symbols
  4. Solve the inequality: Use algebraic techniques
  5. Interpret the solution: Relate back to the real-world context
Variable
\(x = \text{number of books}\)
Inequality
\(8x \leq 50\)
Solve
\(x \leq 6.25\)
Step 1: Define the variable

Let \(x =\) the number of books Sarah can buy

Step 2: Identify the constraint

Sarah can spend AT MOST $50

Cost per book: $8

Total cost: \(8x\)

Step 3: Write the inequality

Since total cost must be less than or equal to budget:

\(8x \leq 50\)

Step 4: Solve the inequality

\(\frac{8x}{8} \leq \frac{50}{8}\)

\(x \leq 6.25\)

Step 5: Interpret the solution

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

\(x \leq 6.25\), so at most 6 books
Final answer:

Sarah can buy at most 6 books (since she can't buy a fraction of a book)

Applied rules:

• Real-world constraints: Solutions must make sense in context

• Inequality symbols: 'at most' means \(\leq\), 'at least' means \(\geq\)

• Integer solutions: Some contexts require whole number answers

Tip: Always define your variable clearly at the beginning!
Tip: Look for key phrases: 'at most', 'no more than', 'at least', 'more than'.
2 Speed and distance
Exercise 2
A car travels at a rate of 60 miles per hour. Write an inequality to represent how long it takes to travel at least 240 miles and solve it.
Definition:

Distance-rate-time relationship: Distance = Rate × Time

'At least': The minimum amount allowed, represented by \(\geq\)

Rate: The speed at which something occurs

Variable
\(t = \text{time in hours}\)
Inequality
\(60t \geq 240\)
Solve
\(t \geq 4\)
Step 1: Define the variable

Let \(t =\) the time in hours

Step 2: Identify the relationship

Distance = Rate × Time

Rate = 60 mph

Time = \(t\) hours

Distance traveled = \(60t\) miles

Step 3: Identify the constraint

The car must travel AT LEAST 240 miles

This means: Distance \(\geq\) 240 miles

Step 4: Write the inequality

\(60t \geq 240\)

Step 5: Solve the inequality

\(\frac{60t}{60} \geq \frac{240}{60}\)

\(t \geq 4\)

Step 6: Interpret the solution

The car must travel for at least 4 hours to cover 240 miles

\(t \geq 4\) hours
Final answer:

The car must travel for at least 4 hours to travel at least 240 miles

Applied rules:

• Distance formula: \(d = rt\) (distance = rate × time)

• Keyword identification: 'at least' means \(\geq\)

• Unit consistency: Make sure units match throughout the problem

Tip: 'At least' means greater than or equal to (\(\geq\))!
Tip: Always check that your units make sense in the context!
3 Academic requirement
Exercise 3
To pass a course, a student must score higher than 75% on the final exam. If the exam has 40 points, write an inequality to represent the minimum number of points needed to pass and solve it.
Definition:

Percentage: A ratio expressed as a fraction of 100

'Higher than': Represented by the symbol \(>\)

Minimum requirement: The lowest acceptable value

Variable
\(x = \text{points scored}\)
Inequality
\(\frac{x}{40} > 0.75\)
Solve
\(x > 30\)
Step 1: Define the variable

Let \(x =\) the number of points scored on the exam

Step 2: Set up the percentage relationship

Score as percentage: \(\frac{x}{40}\)

Required percentage: 75% = 0.75

Step 3: Identify the constraint

The student must score HIGHER THAN 75%

This means: \(\frac{x}{40} > 0.75\)

Step 4: Solve the inequality

\(\frac{x}{40} > 0.75\)

Multiply both sides by 40: \(x > 0.75 \times 40\)

\(x > 30\)

Step 5: Interpret the solution

The student needs more than 30 points to pass

Since points are whole numbers, the student needs at least 31 points

\(x > 30\), so at least 31 points
Final answer:

The student needs more than 30 points to pass, which means at least 31 points

Applied rules:

• Percentage conversion: Divide the part by the whole

• Keyword identification: 'higher than' means \(>\)

• Context interpretation: Round up when dealing with discrete quantities

Tip: 'Higher than' means greater than (\(>\)), not greater than or equal to!
Tip: When dealing with whole items (points, people, etc.), consider integer solutions!
Rules and methods, laws,...
\(\text{Distance} = \text{Rate} \times \text{Time}\)
Distance Formula
\(\text{Percentage} = \frac{\text{Part}}{\text{Whole}}\)
Percentage Formula
At most
\(\leq\)
Maximum allowed
At least
\(\geq\)
Minimum required
More than
\(>\)
Exceeds minimum
Less than
\(<\)
Below maximum
Constraint Property: Word problems involve real-world limitations that create inequalities.
Contextual Solutions: Solutions must make sense in the real-world scenario.
Solution: Exercises 4 to 5
4 Event planning
Exercise 4
A party venue can hold up to 120 people. If 45 people have already confirmed attendance, write an inequality to represent how many more people can attend and solve it.
Definition:

Capacity constraint: A maximum limit on the number of people or items

'Up to': Maximum allowed, represented by \(\leq\)

Remaining capacity: Total capacity minus current usage

Variable
\(x = \text{additional people}\)
Inequality
\(45 + x \leq 120\)
Solve
\(x \leq 75\)
Step 1: Define the variable

Let \(x =\) the number of additional people who can attend

Step 2: Identify the constraint

The venue can hold UP TO 120 people

Current attendees: 45 people

Additional attendees: \(x\) people

Total attendees: \(45 + x\) people

Step 3: Write the inequality

Since total attendees cannot exceed capacity:

\(45 + x \leq 120\)

Step 4: Solve the inequality

\(45 + x - 45 \leq 120 - 45\)

\(x \leq 75\)

Step 5: Interpret the solution

At most 75 more people can attend the party

\(x \leq 75\)
Final answer:

At most 75 more people can attend the party

Applied rules:

• Capacity constraint: Total cannot exceed the maximum limit

• Keyword identification: 'up to' means \(\leq\)

• Additive relationship: Current + Additional = Total

Tip: Capacity problems often involve addition: current + additional = total!
5 Time management
Exercise 5
Maria has 3 hours to complete her homework. She spends 45 minutes on math and plans to spend at least 30 minutes on science. Write an inequality to represent how much time she can spend on English and solve it.
Definition:

Time constraint: A limitation on available time

Unit conversion: Converting between hours and minutes

Minimum requirement: At least 30 minutes for science

Variable
\(x = \text{minutes for English}\)
Total time
\(3 \text{ hours} = 180 \text{ minutes}\)
Inequality
\(45 + 30 + x \leq 180\)
Solve
\(x \leq 105\)
Step 1: Define the variable

Let \(x =\) the number of minutes spent on English homework

Step 2: Convert units to match

Total time available: 3 hours = 180 minutes

Math time: 45 minutes

Science time: at least 30 minutes (minimum)

Step 3: Set up the constraint

Total time spent must be less than or equal to available time

Math + Science + English ≤ Total time

Using minimum for science: \(45 + 30 + x \leq 180\)

Step 4: Solve the inequality

\(75 + x \leq 180\)

\(x \leq 180 - 75\)

\(x \leq 105\)

Step 5: Interpret the solution

Maria can spend at most 105 minutes on English homework

\(x \leq 105\) minutes
Final answer:

Maria can spend at most 105 minutes on English homework

Applied rules:

• Unit consistency: Convert all time measurements to the same unit

• Sum constraint: Individual amounts must sum to no more than the total

• Minimum substitution: Use the minimum required when finding maximum available

Tip: Always convert time units to match before setting up equations!
Tip: When a constraint involves 'at least', use the minimum value to find the maximum available.
Comprehensive Guide to Inequalities Word Problems
\(\text{Constraint: } \text{Quantity} \leq \text{Maximum} \text{ OR } \text{Quantity} \geq \text{Minimum}\)
Constraint Principle
Key definitions:

Word problem: A real-world scenario expressed in words that requires mathematical modeling

Constraint: A limitation or restriction that affects the solution

Variable: The unknown quantity to be determined

Mathematical model: The inequality that represents the real-world situation

Context interpretation: Translating the mathematical solution back to the real-world scenario

Complete methodology:
  1. Read carefully: Understand the situation and identify what's being asked
  2. Define variables: Assign letters to unknown quantities
  3. Identify constraints: Find the limitations and relationships
  4. Translate keywords: Convert phrases to mathematical symbols
  5. Set up inequality: Write the mathematical model
  6. Solve: Use algebraic techniques
  7. Interpret: Relate the solution back to the context
  8. Verify: Check that the solution makes sense
Tip 1: Always define your variable clearly - what does it represent?
Tip 2: Key phrases: 'at most' → ≤, 'at least' → ≥, 'more than' → >, 'less than' → <
Tip 3: Check that your units are consistent throughout the problem!
Tip 4: Consider the real-world context - does your solution make practical sense?
Tip 5: When dealing with discrete quantities (people, items), consider integer solutions!
Common errors: Misinterpreting keywords, inconsistent units, forgetting context, not defining variables clearly.
Exam preparation: Practice various types of constraints (budget, time, capacity), focus on keyword translation, master unit conversions.
Translation rules to know by heart:

• 'At most', 'no more than', 'maximum': ≤

• 'At least', 'no less than', 'minimum': ≥

• 'More than', 'greater than': >

• 'Less than', 'fewer than': <

• 'Up to': ≤

• 'As much as': ≤

• 'As little as': ≥

Word Problem Solving Workflow

📊
Problem-Solving Process
1
Define Variable
2
Identify Constraint
3
Translate Keywords
4
Set Up Inequality
5
Solve
6
Interpret
Keyword Translations
At most → ≤
At least → ≥
More than → >
Less than → <
Up to → ≤
Not exceed → ≤
Must exceed → >
Minimum → ≥
Master Keyword Translations to Excel in Word Problems!

Questions & Answers

Question: How do I know whether to use ≤ or < when a problem says 'at most'?

Answer: 'At most' always means 'less than or equal to' (≤). The phrase explicitly includes the possibility of reaching the maximum value.

Examples:

  • "At most 50 people" means 50 or fewer: \(x \leq 50\)
  • "At most $100" means $100 or less: \(x \leq 100\)
  • "At most 3 hours" means 3 hours or less: \(x \leq 3\)

The word 'at' in 'at most' signals that the exact value is included. If it were just 'less than', that would be different.

Similarly, 'at least' means 'greater than or equal to' (≥).

Question: What if I get a decimal answer for a problem about people or items? Should I round up or down?

Answer: This depends on the context and the direction of the inequality:

For 'at most' problems:

  • If you get \(x \leq 6.25\) for the number of books, you round DOWN to 6
  • You can't exceed the maximum, so you take the largest whole number that satisfies the constraint

For 'at least' problems:

  • If you get \(x \geq 6.25\) for the number of workers needed, you round UP to 7
  • You must meet or exceed the minimum, so you take the smallest whole number that satisfies the constraint

Always consider what makes sense in the real-world context!

Question: How do I handle problems with multiple constraints?

Answer: Problems with multiple constraints result in compound inequalities:

Example: A student must score at least 70% but no more than 95% to earn a B grade.

  • Minimum constraint: Score ≥ 70
  • Maximum constraint: Score ≤ 95
  • Combined: \(70 \leq \text{score} \leq 95\)

You need to satisfy ALL constraints simultaneously. The solution is the intersection of all individual constraint solutions.

For more complex cases, solve each constraint separately, then find the overlap of all solutions.

Remember: The final solution must satisfy every single constraint in the problem!