SAT Math – Algebra
Linear Inequality Word Problems
Translating constraints and ranges into mathematical inequalities
Linear inequality word problems test a critical real-world skill: understanding constraints, limits, and ranges. Unlike equations that have one exact solution, inequalities represent ranges of possible values—budgets you can't exceed, minimum requirements you must meet, maximum capacities that can't be surpassed.
Success requires recognizing key phrases that signal inequality direction, setting up the correct mathematical relationship, solving systematically (watching for sign reversals), and interpreting solutions in context. These problems appear regularly on the SAT and demand both algebraic fluency and careful reading comprehension.
Understanding Linear Inequalities
What Are Linear Inequalities?
A linear inequality is a mathematical statement comparing two expressions using inequality symbols (\(<\), \(>\), \(\leq\), \(\geq\)). While equations give one specific solution, inequalities describe a range of values that satisfy a condition.
• Equation: \(x = 5\) (only 5 works)
• Inequality: \(x > 5\) (5.1, 6, 10, 100... all work)
Inequality Symbols
\(>\) Greater than (does not include the boundary value)
\(<\) Less than (does not include the boundary value)
\(\geq\) Greater than or equal to (includes the boundary value)
\(\leq\) Less than or equal to (includes the boundary value)
Key Phrase Translation Guide
The 5-Step Solution Strategy
Step 1: Identify the Constraint
Read carefully for phrases like "at least," "no more than," "within budget." These signal inequality, not equality.
Step 2: Define Variables and Set Up the Inequality
Assign variables to unknowns. Write the inequality using correct symbols based on the verbal description.
Step 3: Solve the Inequality
Use inverse operations to isolate the variable. Remember: flip the inequality when multiplying or dividing by a negative number!
Step 4: Interpret the Solution
What does the inequality mean in context? If asked for a specific value (minimum, maximum, or possible value), identify it from the solution set.
Step 5: Check Reasonableness
Does your answer make sense? Can you buy -5 items? Does a maximum value of 1000 fit when capacity is 100? Use logic!
The Critical Rule for Inequalities
⚠️ FLIP THE INEQUALITY WHEN MULTIPLYING OR DIVIDING BY A NEGATIVE
Example:
\(-2x < 10\)
Divide both sides by -2 AND flip the inequality:
\(x > -5\) (the \(<\) became \(>\))
This is the #1 mistake on SAT inequality problems!
Common Pitfalls & Expert Tips
❌ Confusing "at least" with "at most"
"At least 10" means 10 or more (\(x \geq 10\)), not less. "At most 10" means 10 or fewer (\(x \leq 10\)). These are opposites!
❌ Using the wrong inequality symbol
"More than 5" is \(x > 5\) (doesn't include 5). "At least 5" is \(x \geq 5\) (includes 5). One word changes the symbol!
❌ Forgetting to flip the inequality
When dividing or multiplying by a negative, the inequality direction reverses. This is non-negotiable and catches many test-takers.
❌ Giving a range when asked for a specific value
If the question asks for "the minimum number," give the smallest value that works, not the entire inequality solution.
✓ Expert Tip: Underline constraint words
Circle or underline "at least," "at most," "no more than" immediately. These tell you exactly which inequality symbol to use.
✓ Expert Tip: Test a value in your solution
After solving, plug in a value from your solution range to verify it satisfies the original inequality. Quick reality check!
✓ Expert Tip: Consider context for whole numbers
If the problem involves discrete items (tickets, people), round appropriately. You can't buy 4.7 tickets—it's either 4 or 5!
Fully Worked SAT-Style Examples
A convenience store requires customers to spend at least $4 to use a debit card. Donuts cost $0.80 each, and a bottle of orange juice costs $2.50. If Ayumi buys one bottle of orange juice, what is the minimum number of donuts she must buy to pay with a debit card?
Solution:
Step 1: Identify the constraint
"At least $4" means the total must be \(\geq 4\)
Step 2: Define variables and set up inequality
Let \(d\) = number of donuts
Cost of donuts: \(0.80d\)
Cost of orange juice: $2.50
Total cost must be at least $4:
\(0.80d + 2.50 \geq 4\)
Step 3: Solve the inequality
\(0.80d + 2.50 \geq 4\)
\(0.80d \geq 1.50\)
\(d \geq \frac{1.50}{0.80} = 1.875\)
Step 4: Interpret for discrete items
Since \(d \geq 1.875\) and you can't buy a fraction of a donut:
The minimum number is 2 donuts (rounding 1.875 up to the next whole number)
Verification:
With 2 donuts: \(0.80(2) + 2.50 = 1.60 + 2.50 = 4.10\)
Is $4.10 at least $4? Yes! ✓
With 1 donut: \(0.80(1) + 2.50 = 3.30\) (not enough) ✗
Answer: Ayumi must buy a minimum of 2 donuts
On a car trip, Rhett and Jessica each drove for part of the trip, and the total distance they drove was under 220 miles. Rhett drove at an average speed of 35 mph, and Jessica drove at an average speed of 40 mph. Which inequality represents this situation, where \(r\) is the number of hours Rhett drove and \(j\) is the number of hours Jessica drove?
Answer choices:
A) \(35r + 40j > 220\)
B) \(35r + 40j < 220\)
C) \(40r + 35j > 220\)
D) \(40r + 35j < 220\)
Solution:
Step 1: Identify the constraint
"Under 220 miles" means less than 220: \(< 220\)
Step 2: Calculate distances
Distance = Speed × Time
Rhett's distance: \(35r\) miles
Jessica's distance: \(40j\) miles
Total distance: \(35r + 40j\)
Step 3: Write the inequality
Total distance is under 220:
\(35r + 40j < 220\)
Step 4: Eliminate wrong answers
A and C use \(>\) instead of \(<\) ✗
D reverses Rhett and Jessica's speeds ✗
B has correct speeds and correct inequality ✓
Answer: B) \(35r + 40j < 220\)
A factory produces widgets at a cost of $5 per unit plus $1,000 in fixed costs. Each widget sells for $8. How many widgets must the factory sell to make a profit? (Profit means revenue exceeds total costs.)
Solution:
Step 1: Define variable and identify constraint
Let \(x\) = number of widgets sold
"Make a profit" means revenue \(>\) total cost
Step 2: Set up expressions
Revenue: \(8x\) dollars
Total cost: \(5x + 1000\) dollars
Profit condition:
\(8x > 5x + 1000\)
Step 3: Solve the inequality
\(8x > 5x + 1000\)
\(3x > 1000\)
\(x > 333.33...\)
Step 4: Interpret for whole units
Since \(x > 333.33\) and widgets are whole units:
Must sell at least 334 widgets to make a profit
Verification:
At 334 widgets:
Revenue: \(8(334) = 2672\)
Cost: \(5(334) + 1000 = 1670 + 1000 = 2670\)
Profit: \(2672 - 2670 = 2\) (yes, profit!) ✓
At 333 widgets: Revenue = 2664, Cost = 2665 (loss of $1) ✗
Answer: The factory must sell at least 334 widgets
A car rental company charges $40 per day plus $0.25 per mile driven. If a customer's budget is $200 for a 3-day rental, what is the maximum number of miles they can drive?
Solution:
Step 1: Define variable and constraint
Let \(m\) = number of miles driven
Budget is $200 means cost \(\leq 200\)
Step 2: Set up the inequality
Daily cost for 3 days: \(40 \times 3 = 120\)
Mileage cost: \(0.25m\)
Total cost:
\(120 + 0.25m \leq 200\)
Step 3: Solve for \(m\)
\(120 + 0.25m \leq 200\)
\(0.25m \leq 80\)
\(m \leq \frac{80}{0.25} = 320\)
Step 4: Identify the maximum
The solution \(m \leq 320\) means the customer can drive up to 320 miles
Maximum number of miles = 320
Verification:
At 320 miles: \(120 + 0.25(320) = 120 + 80 = 200\)
Exactly at budget! ✓
Answer: The maximum is 320 miles
Solve for \(x\): \(-3x + 12 \leq 27\)
Solution:
Step 1: Isolate terms with \(x\)
\(-3x + 12 \leq 27\)
\(-3x \leq 15\)
Step 2: CRITICAL - Dividing by negative!
Divide both sides by -3
FLIP THE INEQUALITY SIGN!
\(x \geq -5\)
The \(\leq\) became \(\geq\)
Verification:
Test \(x = -5\): \(-3(-5) + 12 = 15 + 12 = 27\) ≤ 27 ✓
Test \(x = 0\): \(-3(0) + 12 = 12\) ≤ 27 ✓
Test \(x = -10\): \(-3(-10) + 12 = 30 + 12 = 42\) ≤ 27? No ✗
This confirms \(x \geq -5\) is correct!
Answer: \(x \geq -5\)
Daren has $25 to make international calls. Calls to Japan cost $0.35 per minute, and calls to Turkey cost $0.45 per minute. She wants to call Turkey for at least 30 minutes. Which inequality represents the relationship between \(j\) (minutes calling Japan) and \(t\) (minutes calling Turkey)?
Solution:
Step 1: Identify constraints
Constraint 1: Total cost cannot exceed $25
Constraint 2: Turkey minutes must be at least 30
Step 2: Set up inequalities
Budget constraint:
\(0.35j + 0.45t \leq 25\)
Time constraint:
\(t \geq 30\)
Step 3: This is a system of inequalities
Both conditions must be satisfied:
\(\begin{cases} 0.35j + 0.45t \leq 25 \\ t \geq 30 \end{cases}\)
Finding a possible solution:
If \(t = 30\) (minimum Turkey time):
\(0.35j + 0.45(30) \leq 25\)
\(0.35j + 13.5 \leq 25\)
\(0.35j \leq 11.5\)
\(j \leq 32.86\)
So Daren can call Japan for up to about 32 minutes while calling Turkey for 30 minutes.
Answer: The system is \(0.35j + 0.45t \leq 25\) and \(t \geq 30\)
For snow to form, the temperature must remain below 32°F. If the temperature is currently 45°F and decreasing at a rate of 2°F per hour, after how many hours will it be cold enough for snow?
Solution:
Step 1: Define variable and constraint
Let \(h\) = number of hours from now
"Below 32°F" means temperature \(< 32\)
Step 2: Set up temperature expression
Starting temperature: 45°F
Decreasing 2°F per hour: subtract \(2h\)
Temperature after \(h\) hours: \(45 - 2h\)
Snow condition:
\(45 - 2h < 32\)
Step 3: Solve for \(h\)
\(45 - 2h < 32\)
\(-2h < -13\)
Divide by -2 and FLIP the inequality:
\(h > 6.5\)
Step 4: Interpret
Since \(h > 6.5\), it takes more than 6.5 hours
Minimum time: After 6.5 hours (or practically, after 7 hours)
Verification:
After 6.5 hours: \(45 - 2(6.5) = 45 - 13 = 32\)°F (exactly at freezing)
After 7 hours: \(45 - 2(7) = 31\)°F (below 32) ✓
Answer: After more than 6.5 hours (or at least 7 hours)
Maria needs an average of at least 85 on four tests to earn a B in her class. Her scores on the first three tests are 78, 82, and 91. What is the minimum score she needs on the fourth test to earn a B?
Solution:
Step 1: Define variable and constraint
Let \(x\) = score on fourth test
"At least 85" means average \(\geq 85\)
Step 2: Set up the inequality
Average of four tests:
\(\frac{78 + 82 + 91 + x}{4} \geq 85\)
Simplify:
\(\frac{251 + x}{4} \geq 85\)
Step 3: Solve for \(x\)
\(\frac{251 + x}{4} \geq 85\)
\(251 + x \geq 340\)
\(x \geq 89\)
Verification:
With score of 89: \(\frac{78 + 82 + 91 + 89}{4} = \frac{340}{4} = 85\) ✓
With score of 88: \(\frac{78 + 82 + 91 + 88}{4} = \frac{339}{4} = 84.75\) (not enough) ✗
Answer: Maria needs a minimum score of 89 on the fourth test
Common Question Types
Budget/Cost Problems
• "Within budget" → \(\leq\)
• "Can afford" → \(\leq\)
• "No more than" → \(\leq\)
• Setup: Total cost ≤ Budget
Minimum/Maximum Problems
• "At least" → \(\geq\)
• "Minimum" → smallest value in solution
• "Maximum" → largest value in solution
Profit/Loss Problems
• Profit: Revenue \(>\) Cost
• Loss: Revenue \(<\) Cost
• Break-even: Revenue \(=\) Cost
Average/Mean Problems
• "Average at least" → \(\geq\)
• Setup: \(\frac{\text{sum of values}}{\text{count}} \geq \text{target}\)
SAT Test Day Checklist
✓ Reading Phase
- Underline constraint phrases
- Identify "at least," "at most," "under," "over"
- Note whether asking for min, max, or range
✓ Setup Phase
- Define variables clearly
- Choose correct inequality symbol
- Write complete inequality
✓ Solving Phase
- Isolate variable systematically
- FLIP if dividing/multiplying by negative
- Simplify fully
✓ Answer Phase
- Round appropriately for discrete items
- Extract specific value if asked (min/max)
- Test your answer in original context
Inequalities: The Mathematics of Real-World Constraints
Linear inequality word problems reflect how mathematics operates in reality—where we work within budgets, meet minimum requirements, stay under weight limits, and optimize within constraints. Every business operates on inequalities (profit must exceed costs), every building adheres to inequalities (weight must stay below capacity), and every budget functions as an inequality (spending must not exceed income). The SAT tests these problems because understanding constraints is fundamental to quantitative reasoning in the real world. Master the translation from words to symbols, remember to flip when dividing by negatives, and always verify your solution makes contextual sense. These aren't just test questions—they're life skills.