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📝 Translating sentences to inequalities and solution (14 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 14 questions available

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What is Translating sentences to inequalities and solution?

Definition:
Translating sentences to inequalities involves converting verbal statements into algebraic inequalities using key phrases: 'less than' ( $<$ ), 'at most' ( $\leq$ ), 'greater than' ( $>$ ), 'at least' ( $\geq$ ), and 'between' (compound). Then solve the inequality and interpret the solution.

Working:
For 'A number plus 5 is less than 12', write $x + 5 < 12$ , solve: $x < 7$ . For 'At least 8' means $\geq 8$ .

Example:
Translate 'Twice a number decreased by 3 is at most 11' to $2x - 3 \leq 11$ . Solve: add 3: $2x \leq 14$ , divide by 2: $x \leq 7$ .

Reason:
This skill applies inequalities to real-world contexts like budgeting, speed limits, and minimum requirements.

Easy

Medium

Hard

📝 All Translating sentences to inequalities and solution MCQs

Q1. A student says the sentence 'Seven less than twice a number is at least 15' can be written as $7-2x\ge15$ . Which inequality correctly represents the sentence, and what is its solution?

7-2x\ge15, x\le-4

2x-7\ge15, x\ge11

✅

2x-7\ge15, x\ge-4

7-2x\le15, x\ge-4

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The phrase 'twice a number' is $2x$ , while 'seven less than' means subtract 7 from that quantity, giving $2x-7$ . 'At least 15' means greater than or equal to 15. Solving gives $2x\ge22$ , so $x\ge11$ .

Q2. Which inequality and solution correctly represent the statement 'A number increased by 9 is less than 21'?

x+9>21, x>12

x-9<21, x<30

x+9<21, x<12

✅

9x<21, x<\frac{7}{3}

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: 'Increased by 9' translates to $x+9$ , and 'less than 21' translates to $<21$ . Subtracting 9 from both sides produces $x<12$ . The other choices reverse the inequality, subtract instead of add, or incorrectly interpret multiplication.

Q3. A school club has a budget of Rs. 4,500. It has already spent Rs. 1,250 and plans to spend Rs. 375 on each remaining activity. What inequality models the situation, and how many activities can still be funded?

1250+375x\le4500, x\le8

✅

1250+375x\ge4500, x\ge8

375x-1250\le4500, x\le15

1250-375x\le4500, x\ge-8

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The total spending consists of the Rs. 1,250 already spent plus Rs. 375 for each activity, so $1250+375x\le4500$ . Subtracting 1,250 gives $375x\le3250$ , hence $x\le8.66$ . Because activities are whole numbers, at most 8 activities can be funded.

Q4. A taxi charges Rs. 200 initially plus Rs. 35 per kilometer. A passenger can spend no more than Rs. 900. Which inequality correctly models the maximum distance $d$ , and what is the greatest whole number of kilometers possible?

200+35d\ge900, d\ge20

200+35d\le900, d\le20

✅

35d-200\le900, d\le31

200-35d\le900, d\ge20

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: 'No more than Rs. 900' means the total cost must be less than or equal to 900. Thus $200+35d\le900$ . Subtracting 200 gives $35d\le700$ , and dividing by 35 gives $d\le20$ . Therefore, 20 kilometers is the greatest possible whole-number distance.

Q5. A teacher translates 'Five times a number decreased by 8 is greater than 27' as $5(x-8)>27$ . What is the best analysis of the teacher's translation?

A.It is correct because 'decreased by 8' always applies to the entire expression.

B.It is incorrect because the correct translation is

5x-8>27

. ✅

C.It is incorrect because the correct translation is

5x+8>27

D.It is correct, but its solution must be

x>35

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The wording 'five times a number decreased by 8' means first multiply the number by 5 and then subtract 8, giving $5x-8>27$ . Writing $5(x-8)$ would instead mean five times the quantity obtained after subtracting 8 from the number.

Q6. A student solves 'Three less than four times a number is at most 17' as $3-4x\le17$ , obtaining $x\ge-3.5$ . What mistake did the student make, and what is the correct solution?

A.They reversed 'at most'; the correct solution is

x\ge5

B.They reversed the order in 'three less than'; the correct solution is

x\le5

. ✅

C.They should multiply 3 by 4; the correct solution is

x\le\frac{17}{12}

D.They should use equality only; the correct solution is

x=5

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: 'Three less than four times a number' means $4x-3$ , not $3-4x$ . Therefore $4x-3\le17$ , so $4x\le20$ , giving $x\le5$ . The student's error changes the expression before solving, producing a completely different inequality.

Q7. A gym charges a membership fee of Rs. 1,000 plus Rs. 150 for each class attended. A member wants the total cost to be strictly below Rs. 2,800. Which solution correctly describes the number $c$ of classes?

1000+150c<2800, c<12

✅

1000+150c\le2800, c\le12

150c-1000<2800, c<25

1000+150c>2800, c>12

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: 'Strictly below' requires a strict inequality, $<$ , rather than $\le$ . The model is $1000+150c<2800$ . Subtracting 1000 gives $150c<1800$ , and division by 150 gives $c<12$ . Thus 11 is the greatest whole-number class count.

Q8. A student interprets 'The difference between a number and 6 is no greater than 10' as $x-6>10$ . Which correction is appropriate?

A.Use

x-6\ge10

, because 'no greater' means greater than or equal to.

B.Use

x-6\le10

, because 'no greater' means less than or equal to. ✅

C.Use

6-x\le10

, because difference always reverses the order.

D.Use

x+6\le10

, because subtracting is not allowed in an inequality.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The phrase 'no greater than' means 'less than or equal to.' Therefore the correct translation is $x-6\le10$ . Adding 6 to both sides gives $x\le16$ . The key issue is interpreting the comparison phrase correctly before performing algebraic operations.

Q9. A number line shows a closed circle at 4 with shading extending to the left. Which sentence and solution match the graph?

A.The number is greater than 4, so

x>4

B.The number is at least 4, so

x\ge4

C.The number is no greater than 4, so

x\le4

. ✅

D.The number is less than 4, so

x<4

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A closed circle means the endpoint is included, while shading to the left represents values smaller than the endpoint. Therefore the graph represents $x\le4$ , which can be described as the number being no greater than 4.

Q10. Two students solve 'Twice a number plus 5 is at least 19.' Student A writes $2x+5\ge19$ and obtains $x\ge7$ . Student B writes $2(x+5)\ge19$ and obtains $x\ge4.5$ . Which conclusion is correct?

A.Student A is correct because 'twice a number' applies only to the number. ✅

B.Student B is correct because 'plus 5' must always be inside parentheses.

C.Both are correct because they represent equivalent expressions.

D.Neither is correct because 'at least' means

<

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The phrase 'twice a number plus 5' translates directly to $2x+5\ge19$ . Solving gives $2x\ge14$ , hence $x\ge7$ . Student B incorrectly multiplies the 5 by 2, changing the original meaning of the sentence.

Q11. A number is increased by 40% and the result must be no more than 84. If the original number is $x$ , which inequality and solution are correct?

x+0.4\le84, x\le83.6

1.4x\le84, x\le60

✅

0.4x\le84, x\le210

1.4x\ge84, x\ge60

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Increasing a number by 40% means adding $0.4x$ to the original $x$ , producing $1.4x$ . 'No more than 84' gives $1.4x\le84$ . Dividing by 1.4 yields $x\le60$ , so 60 is the greatest possible original value.

Q12. A graph represents all values $x$ satisfying $x\ge-2$ . Which verbal statement describes the same solution?

A.A number is less than negative 2.

B.A number is no more than negative 2.

C.A number is at least negative 2. ✅

D.A number is greater than negative 2.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The symbol $x\ge-2$ means that $x$ is greater than or equal to negative 2. The phrase 'at least' precisely expresses greater than or equal to, while 'greater than' alone would exclude $-2$ .

Q13. A delivery service charges a fixed Rs. 500 plus Rs. 80 per package. A customer wants to keep the bill below Rs. 2,100. After solving the inequality, the customer claims 20 packages are affordable. Is the claim valid?

A.Yes, because

500+80(20)=2100

, which is below the limit.

B.Yes, because 20 is less than 21 packages.

C.No, because the inequality gives

x<20

, so the greatest whole number is 19. ✅

D.No, because the inequality gives

x>20

, so at least 21 packages are required.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The model is $500+80x<2100$ because the bill must be strictly below 2,100. Subtracting 500 gives $80x<1600$ , so $x<20$ . Although 20 gives exactly Rs. 2,100, equality is not allowed; therefore 19 is the greatest whole-number choice.

Q14. For real numbers $x$ , the statement 'Three times a number is more than 8 less than twice the same number' can be translated and solved. Which result is correct?

3x>2x-8, x>-8

✅

3x>2x+8, x>8

3x-8>2x, x>8

3x<2x-8, x<-8

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: 'Eight less than twice the number' is $2x-8$ , so the comparison is $3x>2x-8$ . Subtracting $2x$ from both sides gives $x>-8$ . This requires careful interpretation of both the comparison phrase and the order of subtraction.

🔗 Related Topics (MCQs)

📝 Subtraction and Addition Properties of Equality MCQs 📝 How to verify a solution to an equation MCQs 📝 Subtraction Property of Equality MCQs 📝 Addition Property of Equality MCQs 📝 Solving equations with simplification MCQs

←Previous Topic:Solve inequalities that require simplification📝 Translating sentences to inequalities and solutionClassify inequalities as identities or contradictions:Next Topic→