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📝 At least inequality meaning and examples (10 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 10 questions available

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What is At least inequality meaning and examples?

Definition:
At least" in inequality problems means the quantity is greater than or equal to a certain value represented by the symbol $\ge$ and it indicates the minimum acceptable amount meaning the value cannot fall below the stated figure and this is used for requirements like minimum score minimum sales or minimum time.

Working:
To translate "at least use $\ge$ and set the variable on the left side; for example, at least 50" translates to $x \ge 50$ and when solving the solution includes the boundary value and all values greater than it and this is crucial for meeting minimum criteria and graphs show a closed circle at the boundary with an arrow to the right.

Example:
To pass a course a student needs an average of at least 70% on four exams; if the first three scores are 6572 and 68 and the fourth is $x$ then $\frac{65 + 72 + 68 + x}{4} \ge 70$ so $205 + x \ge 280$ $x \ge 75$ meaning the fourth exam score must be at least 75%.

Reason:
Understanding "at least" is important for setting goals meeting standards and ensuring thresholds are reached and it applies in education manufacturing and finance where minimum requirements are common."

Easy

Medium

Hard

📝 All At least inequality meaning and examples MCQs

Q1. A school requires students to collect at least 18 service hours. If a student has already completed 11 hours, which inequality correctly represents the additional hours $x$ needed?

x\le7

x\ge7

✅

x>29

x\le29

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The student needs enough additional hours to reach a total of at least 18. Since $11+x\ge18$ , subtracting 11 from both sides gives $x\ge7$ . The phrase 'at least' includes the boundary value, so exactly 7 hours is acceptable.

Q2. Which inequality best represents the statement: 'A package must weigh at least 5 kilograms but cannot exceed 12 kilograms'?

5<x<12

5\le x\le12

✅

5<x\le12

5\le x<12

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: 'At least 5' means the weight may equal 5, while 'cannot exceed 12' means it may equal 12. Therefore both endpoints are included, giving $5\le x\le12$ . Confusing 'at least' with 'greater than' would incorrectly exclude 5.

Q3. A charity wants to raise at least \ $2,400. It has already collected \$ 1,575. If each additional donor contributes \$75, what is the minimum number of additional donors required?

A.9

B.10 ✅

C.11

D.12

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Let $d$ represent the number of additional donors. The model is $1575+75d\ge2400$ . Subtracting gives $75d\ge825$ , so $d\ge11$ . Therefore the minimum is actually 11 donors; the correct choice is C. This also illustrates why the inequality must be solved rather than estimated.

Q4. A student solves $4x+3\ge19$ and writes $x\ge4$ . Which statement best evaluates the solution?

A.It is correct because 4 makes both sides equal. ✅

B.It is incorrect because the inequality sign must reverse.

C.It is incorrect because

x\ge4

should be

x\ge5

D.It is incorrect because the constant should be subtracted from both sides twice.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Subtracting 3 gives $4x\ge16$ , and dividing by the positive number 4 gives $x\ge4$ . Because equality is allowed, $x=4$ satisfies the original inequality exactly. The inequality sign does not reverse when dividing by a positive number.

Q5. A delivery company charges \ $8 per package plus a \$ 20 service fee. A customer has a budget of \$100 and wants to send at least 7 packages. Which conclusion is correct?

A.The customer can send at most 10 packages and meets the requirement.

B.The customer can send exactly 7 packages but cannot send 8.

C.The customer can send at least 7 packages and at most 10 packages. ✅

D.The customer cannot meet the requirement because the fee is fixed.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The cost condition is $20+8x\le100$ , giving $x\le10$ . The requirement is $x\ge7$ . Combining them produces $7\le x\le10$ , so 7, 8, 9, or 10 packages are possible. The key is combining a minimum requirement with a maximum budget.

Q6. A graph of a solution set on a number line shows a closed circle at 6 with shading extending to the right. Which inequality does the graph represent?

x>6

x\ge6

✅

x<6

x\le6

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A closed circle indicates that the endpoint is included, while shading to the right represents values greater than the endpoint. Therefore the graph represents $x\ge6$ . A common misconception is to associate any circle at 6 with strict inequality without checking whether it is open or closed.

Q7. A theater must sell at least 450 tickets. Adult tickets cost \ $12 and student tickets cost \$ 8. If 20 adult tickets are sold, which inequality determines the minimum number $s$ of student tickets needed to generate at least \$4,000 in revenue?

240+8s\ge4000

✅

240+8s\le4000

12(20+s)\ge4000

20+8s\ge4000

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Twenty adult tickets produce $20(12)=240$ dollars. If $s$ student tickets are sold, their revenue is $8s$ . Requiring total revenue of at least \$4,000 gives $240+8s\ge4000$ . The ticket-count requirement is separate and should not be inserted into the revenue expression.

Q8. A student claims that solving $-3x\ge12$ gives $x\ge-4$ . What is the correct assessment?

A.Correct, because 12 divided by 3 is 4.

B.Correct, because negative coefficients never affect inequality direction.

C.Incorrect; the solution is

x\le-4

. ✅

D.Incorrect; the solution is

x\le4

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Dividing an inequality by a negative number reverses its direction. Thus $-3x\ge12$ becomes $x\le-4$ . The student's mistake is keeping the original direction despite dividing by $-3$ . Testing $x=-5$ confirms the corrected result because $15\ge12$ .

Q9. A graph represents all $x$ -values from 3 through 9, with both endpoints included. Which situation could this graph model?

A.A temperature must be greater than 3 and less than 9 degrees.

B.A worker must complete at least 3 but no more than 9 tasks. ✅

C.A score must be below 3 or above 9.

D.A quantity must be exactly 3 or exactly 9.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The graph corresponds to $3\le x\le9$ , meaning every value from 3 through 9 is allowed. A requirement of at least 3 tasks gives $x\ge3$ , while no more than 9 gives $x\le9$ . Combining them produces the interval shown.

Q10. A student models a savings goal with $50+15w\ge200$ , where $w$ is the number of weeks. Another student says the answer must be $w\ge13.33$ , so 13 weeks is sufficient. Which evaluation is most accurate?

A.The second student is correct because 13.33 rounds down to 13.

B.The second student is incorrect; 13 weeks gives less than \$200, so 14 weeks are required. ✅

C.The second student is correct because savings can be fractional.

D.Both students are incorrect because the inequality should be

w\le13.33

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Solving gives $15w\ge150$ , so $w\ge10$ , not 13.33. Therefore the proposed calculation itself is wrong. At 10 weeks, the savings are exactly \$200. This question tests whether a student verifies the algebra rather than accepting an apparently reasonable numerical claim.

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