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📝 How to Use interval notation (15 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 15 questions available

←Previous Topic:Graph inequalities on the number line📝 How to Use interval notationSolve inequalities using the Subtraction Property of Inequality:Next Topic→

What is How to Use interval notation?

Definition:
Interval notation is a concise way to represent the solution set of an inequality using brackets and parentheses. Square brackets $[ \, ]$ indicate inclusion of an endpoint ( $\leq$ or $\geq$ ), while parentheses $( \, )$ indicate exclusion ( $<$ or $>$ ). Infinity ( $\infty$ ) always uses a parenthesis.

Working:
For $x > 2$ , write $(2, \infty)$ . For $x \leq 5$ , write $(-\infty, 5]$ . For $-3 < x \leq 4$ , write $(-3, 4]$ .

Example:
If solution is $y < -1$ or $y \geq 3$ , interval notation is $(-\infty, -1) \cup [3, \infty)$ .

Reason:
Interval notation is compact and widely used in higher mathematics to describe solution sets efficiently.

Easy

Medium

Hard

📝 All How to Use interval notation MCQs

Q1. Which interval notation correctly represents all real numbers greater than -3 but less than or equal to 5?

[-3,5]

(-3,5]

✅

[-3,5)

(-3,5)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The phrase greater than $-3$ excludes $-3$ , so a parenthesis is required there. The phrase less than or equal to 5 includes 5, so a bracket is required there. Therefore, $(-3,5]$ is correct.

Q2. A student writes $x\geq2$ as $(2,\infty)$ . What is the most important error in the student's notation?

A.The infinity symbol should be replaced by a bracket

B.The endpoint 2 should use a bracket because it is included ✅

C.The interval must begin at 0

D.The infinity symbol must use a bracket

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Because $x\geq2$ includes 2, the endpoint must be represented with a bracket. The infinity symbol always uses a parenthesis because infinity is not an actual real number that can be included.

Q3. A temperature-control system allows temperatures from 18°C through 24°C, including both endpoints. Which interval represents the allowed temperatures?

(18,24)

[18,24)

(18,24]

[18,24]

✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The wording through 24°C and from 18°C indicates that both endpoint temperatures are permitted. Included endpoints require brackets, so the correct interval is $[18,24]$ .

Q4. A solution is described as all values $x<-1$ . Which interval notation correctly captures the solution?

(-\infty,-1)

✅

(-\infty,-1]

[-1,\infty)

(-1,\infty)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The inequality $x<-1$ contains every number to the left of $-1$ , but it does not include $-1$ . Thus a parenthesis is required at $-1$ , and infinity also requires a parenthesis.

Q5. A rental company charges a special rate when the rental duration $t$ satisfies $2<t\leq7$ days. Which interval should be used to represent the eligible durations?

[2,7]

(2,7]

✅

[2,7)

(2,7)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The duration must be greater than 2, so 2 is excluded and receives a parenthesis. Since $t\leq7$ , 7 is included and receives a bracket. Therefore, $(2,7]$ is correct.

Q6. A number-line graph has an open circle at -4 and is shaded to the right. Which interval notation matches the graph?

[-4,\infty)

(-4,\infty)

✅

(-\infty,-4)

(-\infty,-4]

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: An open circle means the endpoint is excluded, while shading to the right represents numbers greater than the endpoint. Therefore, the graph represents $x>-4$ , written as $(-4,\infty)$ .

Q7. A number-line graph shows shading from -2 to 6, with a closed circle at -2 and an open circle at 6. What interval is represented?

(-2,6)

[-2,6]

[-2,6)

✅

(-2,6]

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The closed circle at -2 means -2 is included, while the open circle at 6 means 6 is excluded. Shading between them gives the interval $[-2,6)$ .

Q8. A student converts $x\leq3$ into $(-\infty,3)$ . Another converts it into $(-\infty,3]$ . Which reasoning best explains why the second conversion is correct?

A.Three is included, so a bracket is required ✅

B.Infinity must be written with a bracket

C.The inequality points toward positive numbers

D.A parenthesis is always required for finite numbers

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The symbol $\leq$ means that 3 is part of the solution set. Therefore, the endpoint 3 must use a bracket. The interval extends indefinitely left, so infinity uses a parenthesis.

Q9. A machine operates safely when its pressure $p$ is at least 40 psi but below 75 psi. Which interval represents the safe pressure range?

[40,75)

✅

(40,75]

[40,75]

(40,75)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: At least 40 means 40 is included, requiring a bracket. Below 75 means 75 is excluded, requiring a parenthesis. Hence the safe range is $[40,75)$ .

Q10. Which interval represents $x<-5$ or $x\geq2$ ?

(-\infty,-5) \cup [2,\infty)

✅

(-\infty,-5] \cup (2,\infty)

[-5,2)

(-5,2]

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The first inequality contains numbers strictly less than -5, giving $(-\infty,-5)$ . The second contains 2 and everything greater, giving $[2,\infty)$ . Since the regions are separate, union notation combines them.

Q11. A student claims that $[-3,4]$ represents $x>-3$ and $x<4$ . Which correction is necessary?

A.Both endpoints should be parentheses ✅

B.Only -3 should use a parenthesis

C.Only 4 should use a parenthesis

D.The interval is correct as written

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Both inequalities are strict: $x>-3$ excludes -3, and $x<4$ excludes 4. Therefore, both endpoints require parentheses, giving $(-3,4)$ , not $[-3,4]$ .

Q12. A scholarship requires a test score $s$ of at least 70 but less than 90. A second program accepts scores from 85 through 100, including 100. Which interval represents scores eligible for either program?

[70,90) \cup [85,100]

✅

[70,85) \cup (90,100]

(70,90] \cup [85,100)

[70,90] \cup (85,100]

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The first program accepts $[70,90)$ , while the second accepts $[85,100]$ . Because students can qualify for either program, the union of the two intervals represents the complete eligible set. Overlap is allowed.

Q13. A graph represents $x\geq-1$ . A student writes $[-1,\infty]$ . What single notation rule makes the student's answer invalid?

A.The finite endpoint cannot use a bracket

B.The infinity endpoint must use a parenthesis ✅

C.Negative endpoints cannot use brackets

D.Intervals cannot contain infinity

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The value $-1$ is included, so its bracket is correct. However, infinity is not a real endpoint and cannot be included in an interval. Infinity must always be paired with a parenthesis, giving $[-1,\infty)$ .

Q14. A delivery service accepts package weights greater than 2 kg and no more than 10 kg. A customer argues that $[2,10)$ is correct because both numbers mark boundaries. What is the best evaluation?

A.Correct, because boundaries always use brackets

B.Correct, because 2 and 10 are both possible weights

C.Incorrect, because 2 is excluded and 10 is included ✅

D.Incorrect, because both endpoints must be excluded

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The phrase greater than 2 excludes 2, requiring a parenthesis. The phrase no more than 10 includes 10, requiring a bracket. Thus the correct interval is $(2,10]$ , not $[2,10)$ .

Q15. Which interval notation is equivalent to all real numbers except $x=4$ ?

(-\infty,4) \cup (4,\infty)

✅

(-\infty,4] \cup [4,\infty)

[-\infty,4) \cup (4,\infty]

(-\infty,4) \cup [4,\infty)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: All real numbers less than 4 form $(-\infty,4)$ , while all real numbers greater than 4 form $(4,\infty)$ . Because 4 itself is excluded, both intervals use parentheses at that point, and union combines them.

🔗 Related Topics (MCQs)

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←Previous Topic:Graph inequalities on the number line📝 How to Use interval notationSolve inequalities using the Subtraction Property of Inequality:Next Topic→