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📝 Graph inequalities on the number line (13 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 13 questions available

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What is Graph inequalities on the number line?

Definition:
Graphing inequalities on the number line involves representing the solution set of an inequality visually. Use an open circle for strict inequalities ( $<$ or $>$ ) and a closed circle for inclusive inequalities ( $\leq$ or $\geq$ ). Then shade the region in the direction of the solution.

Working:
For $x > 3$ , place an open circle at 3 and shade to the right. For $x \leq -2$ , place a closed circle at -2 and shade to the left.

Example:
Graph $y \geq 1$ . Place a closed circle at 1 (since included) and shade to the right, indicating all numbers greater than or equal to 1.

Reason:
Number line graphs provide a clear visual representation of the solution set, making it easier to understand the range of values that satisfy the inequality.

Easy

Medium

Hard

📝 All Graph inequalities on the number line MCQs

Q1. Which number-line description correctly represents $x>-3$ ?

A.Closed circle at -3 with shading left

B.Open circle at -3 with shading right ✅

C.Closed circle at -3 with shading right

D.Open circle at -3 with shading left

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The symbol $>$ means that $-3$ itself is excluded, so the endpoint must be an open circle. Because values greater than $-3$ lie to the right, the ray extends rightward.

Q2. A student wants to graph $x\leq5$ . Which combination of endpoint and direction is necessary?

A.Open circle at 5, shading right

B.Closed circle at 5, shading right

C.Open circle at 5, shading left

D.Closed circle at 5, shading left ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The symbol $\leq$ includes the boundary value 5, requiring a closed circle. Since the solution contains numbers less than 5, all represented values extend to the left.

Q3. Two students graph $x\geq2$ . Student A uses an open circle at 2 and shades right. Student B uses a closed circle at 2 and shades right. Who is correct and why?

A.Student A, because greater values exclude 2

B.Student B, because 2 is included ✅

C.Both, because endpoint type does not matter

D.Neither, because the graph must shade left

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Student B is correct because the inequality includes equality. The value 2 satisfies $x\geq2$ , so the point must be closed, while numbers greater than 2 are represented to the right.

Q4. A number line has an open circle at -1 and shading toward larger numbers. Which inequality does this graph represent?

x<-1

x\leq-1

x>-1

✅

x\geq-1

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: An open circle means $-1$ is excluded, while shading toward larger values means the solution contains numbers greater than $-1$ . Therefore the graph corresponds exactly to $x>-1$ .

Q5. A temperature warning system activates when the temperature is at least 80°F. Which number-line graph correctly models the temperatures that activate the warning?

A.Open circle at 80, shading left

B.Closed circle at 80, shading left

C.Open circle at 80, shading right

D.Closed circle at 80, shading right ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: At least 80°F means $T\geq80$ . The temperature 80 activates the warning, so a closed circle is required. Higher temperatures lie to the right and must also be included.

Q6. A parking garage charges a penalty for parking longer than 3 hours. Which graph represents the parking times that result in a penalty?

A.Closed circle at 3, shading left

B.Open circle at 3, shading left

C.Closed circle at 3, shading right

D.Open circle at 3, shading right ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The phrase longer than 3 hours excludes exactly 3 hours, requiring an open circle. Times exceeding 3 hours are greater than 3, so the solution extends toward the right.

Q7. A student solves $2x-6>4$ and obtains $x>5$ . Which number-line graph should the student use?

A.Closed circle at 5, shading left

B.Open circle at 5, shading left

C.Closed circle at 5, shading right

D.Open circle at 5, shading right ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Adding 6 gives $2x>10$ , and dividing by positive 2 gives $x>5$ . Since 5 is excluded, use an open circle, and greater values are shown to the right.

Q8. A delivery service accepts packages weighing no more than 20 kg. A student graphs an open circle at 20 and shades left. What is the best correction?

A.Keep the graph because 20 kg is excluded

B.Change the circle to closed because 20 kg is allowed ✅

C.Move the circle to 21 because 20 kg is too heavy

D.Shade right because lighter packages are larger

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: No more than 20 kg means the weight can equal 20 kg as well as be smaller. Therefore the inequality is $w\leq20$ , which requires a closed circle at 20 and shading to the left.

Q9. Solve $3-2x\leq9$ , then identify its graph. Which choice is correct?

x\leq-3

: closed circle at -3, shading left

x\geq-3

: closed circle at -3, shading right ✅

x\leq3

: closed circle at 3, shading left

x\geq3

: open circle at 3, shading right

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Subtracting 3 gives $-2x\leq6$ . Dividing by $-2$ reverses the inequality, producing $x\geq-3$ . Because equality is included, the endpoint is closed and the ray extends right.

Q10. A number line shows a closed circle at 4 with shading to the left. Another graph shows an open circle at 4 with shading to the right. Which statement compares their solution sets correctly?

A.The first represents

x<4

, and the second represents

x\geq4

B.The first represents

x\leq4

, and the second represents

x>4

✅

C.Both represent

x=4

D.Both represent all real numbers except 4

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A closed circle includes the endpoint, and leftward shading represents smaller values, giving $x\leq4$ . An open circle excludes 4, while rightward shading represents larger values, giving $x>4$ .

Q11. A factory accepts items whose lengths are greater than 12 cm but less than or equal to 18 cm. Which number-line representation is correct?

A.Closed circles at 12 and 18 with shading between them

B.Open circles at 12 and 18 with shading between them

C.Open circle at 12, closed circle at 18, shading between them ✅

D.Closed circle at 12, open circle at 18, shading between them

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The condition combines $x>12$ and $x\leq18$ . Thus 12 is excluded and needs an open circle, while 18 is included and needs a closed circle. Only values between them qualify.

Q12. A number line graph has an open circle at 7 and shading left. A student claims it represents $x\geq7$ . What is the most precise evaluation?

A.Correct, because open circles indicate values greater than or equal to the endpoint

B.Incorrect, because the graph represents

x<7

✅

C.Correct, because shading left always represents greater values

D.Incorrect, because the graph represents

x\leq7

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The open circle excludes 7, while shading left represents values less than 7. Therefore the graph represents $x<7$ , not $x\geq7$ . Both the endpoint and direction contradict the student's claim.

Q13. A competition requires a score $s$ satisfying $s>60$ and $s\leq90$ . Which description correctly represents the complete solution on a number line?

A.Closed circles at 60 and 90, with shading outside the interval

B.Open circle at 60, closed circle at 90, with shading between them ✅

C.Closed circle at 60, open circle at 90, with shading between them

D.Open circles at both endpoints, with shading between them

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The two conditions must hold simultaneously, so scores must lie between 60 and 90. Because 60 is excluded, its endpoint is open; because 90 is included, its endpoint is closed.

🔗 Related Topics (MCQs)

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