Number line

Correct answer: A. 14

Explanation: The distance between two numbers on a number line is the absolute value of their difference: \(\left|-15-(-1)\right|=|-14|=14\). Therefore, the correct answer is 14. A value such as 13 results from incorrect subtraction. Exam tip: a distance is always non-negative.

Correct answer: A. The student's conclusion is correct; \(\sqrt{7}\) lies between 2 and 3 and is closer to 3.

Explanation: Since \(2^2=4\) and \(3^2=9\), \(\sqrt{7}\) lies between 2 and 3. Also, \(\sqrt{7}\approx2.646\), so its distance from 3 is about 0.354, less than 0.646 from 2. Thus option B is wrong. Exam tip: compare nearby perfect squares first.

Correct answer: A. -12

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-6+(-18)}{2}=\frac{-24}{2}=-12\). Therefore, the correct answer is \(-12\). Although \(-11\) may seem close, it is not equally distant from both numbers. Exam tip: include the signs of negative numbers while adding them for a midpoint.

Correct answer: A. 60

Explanation: The distance between two numbers on a number line is the absolute value of their difference. Thus, \(|10-(-50)|=|60|=60\). Although \(-60\) can arise as a signed difference in the reverse order, a distance can never be negative. Exam tip: always take the absolute value of the difference when finding distance.

Correct answer: B. -0.00201

Explanation: For negative numbers, the number closer to zero is greater. Here, \(-0.00201 > -0.00210\) because \(-0.00201\) is closer to zero. Writing \(-0.0021\) as \(-0.00210\) makes the comparison clear. Exam tip: Add trailing zeros when needed to compare the same number of decimal places.

Correct answer: A. \(8\)

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-4+20}{2}=\frac{16}{2}=8\). Therefore, the correct answer is \(8\). \(10\) would be the midpoint of 0 and 20, so it is not correct here. Exam tip: add the two numbers and divide by 2 to find their midpoint.

Correct answer: A. 20

Explanation: On a number line, the distance between two numbers is the absolute value of their difference: \(\lvert -25-(-5)\rvert=\lvert -20\rvert=20\). Therefore, the correct answer is 20. A value such as 15 can result from incorrect subtraction. Exam tip: distance is always non-negative.

Correct answer: B. \(-0.506\)

Explanation: Write \(-0.56\) as \(-0.560\) to compare the decimals easily. Since \(-0.506\) is closer to zero than \(-0.560\), it is greater; among negative numbers, the number closer to zero is greater. Therefore, \(-0.506\) is the correct answer. Exam tip: Make the number of decimal places equal before comparing decimals.

Correct answer: A. \(-2\)

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-9+5}{2}=\frac{-4}{2}=-2\). Therefore, \(-2\) is correct. \(-1\) is not correct because it is not at an equal distance from \(-9\) and \(5\). Exam tip: Add the two numbers and divide by 2 to find their midpoint.

Correct answer: A. 8

Explanation: On a number line, the distance between two numbers is the absolute value of their difference: \(|-12-(-20)|=|-12+20|=|8|=8\). Therefore, the correct answer is 8. A value such as 7 can result from an incorrect subtraction, and distance can never be negative. Exam tip: always use \(|a-b|\) to find the distance between two numbers.

Correct answer: B. -0.71

Explanation: For comparison, write -0.71 as -0.710. Between -0.710 and -0.701, -0.710 is more negative, so it lies further left on the number line and is smaller. Therefore, -0.71 is correct. -0.701 is greater because it is closer to zero. Exam tip: Among negative decimals, the number with the greater magnitude is smaller.

Correct answer: A. \(-10\)

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-3+(-17)}{2}=\frac{-20}{2}=-10\). Therefore, \(-10\) is correct. \(-9\) is not correct because it is not equally distant from \(-3\) and \(-17\). Exam tip: when adding negative numbers, add their magnitudes and keep the negative sign.

Correct answer: A. 30

Explanation: The distance between two numbers on a number line is the absolute value of their difference: \(\left|-45-(-15)\right|=\left|-30\right|=30\). Therefore, the correct answer is 30. \(-30\) can be the signed difference, but distance can never be negative. Exam tip: always take the absolute value when finding distance.

Correct answer: B. Infinitely many rational numbers

Explanation: The correct answer is infinitely many rational numbers. The midpoint \(\frac{a+b}{2}\) is rational and lies between \(a\) and \(b\). Repeatedly taking midpoints produces endlessly many rational numbers. Exam tip: between any two distinct rational numbers, rationals are always infinite.

Correct answer: A. 30

Explanation: On a number line, the distance between two numbers is the absolute value of their difference. Thus, \(\lvert -60-(-30)\rvert=\lvert -30\rvert=30\). Therefore, the correct answer is 30. \(-30\) is the signed difference, not the distance, because distance cannot be negative. Exam tip: always use the absolute value of the difference when finding distance.

Correct answer: B. \(-1\)

Explanation: For negative numbers, the number closer to zero is greater. Since \(-1\) is closer to zero than \(-1.1\), \(-1 > -1.1\). Option \(-1.1\) is smaller because it lies further left on the number line. Exam tip: While comparing negative decimals, remember that a larger magnitude means a smaller number.

Correct answer: B. ( -0.06 )

Explanation: -0.06 is greater than -0.6 because it lies closer to zero and to the right on the number line. In fact, -0.6 = -0.60, and -0.06 > -0.60. Option A is more negative, so it is smaller. Exam tip: when comparing negative decimals, the number closer to zero is greater.

Correct answer: A. 1

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-11+13}{2}=\frac{2}{2}=1\). Therefore, 1 is correct. Zero is not equally distant from the two numbers: it is 11 units from -11 and 13 units from 13. Exam tip: add the two numbers and divide by 2 to find their midpoint.

Correct answer: A. 12

Explanation: On a number line, the distance between two numbers is the absolute value of their difference: \(\lvert -7-(-19)\rvert=\lvert 12\rvert=12\). Therefore, the correct answer is 12. Values such as 11 or 13 result from subtracting incorrectly by one; they are not the actual difference. Exam tip: always express distance as a non-negative value.

Correct answer: B. ( -0.108 )

Explanation: -0.108 is greater because it is closer to zero than -0.18. Writing -0.18 as -0.180 makes the comparison clear: -0.108 > -0.180. Option A is smaller, not greater. Exam tip: for negative decimals, the number with the smaller absolute value is greater.

Correct answer: A. 6

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-4+16}{2}=\frac{12}{2}=6\). Therefore, 6 is correct. Neither 5 nor 7 is at an equal distance from -4 and 16. Exam tip: add the two numbers and divide by 2 to find their midpoint.

Correct answer: A. \(\sqrt{5}\)

Explanation: Since 5 is not a perfect square, \(\sqrt{5}\) is irrational and cannot be written as \(p/q\). Steps of \(1/n\) reach only rational points of the form \(k/n\). Exam tip: terminating and repeating decimals are rational.

Correct answer: A. 14

Explanation: On a number line, the distance between two numbers is the absolute value of their difference. Thus, \(|-13-(-27)|=|-13+27|=|14|=14\). Therefore, 14 is correct. The value 13 is only the absolute value of one number, not the distance between the two numbers. Exam tip: A distance is always non-negative.

Correct answer: A. -6

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-2+(-10)}{2}=\frac{-12}{2}=-6\). Therefore, \(-6\) is correct. \(-5\) is not the midpoint because it is not equally distant from \(-2\) and \(-10\). Exam tip: add the two numbers and divide by 2 to find their midpoint.

Correct answer: A. Point A lies to the left of B.

Explanation: Since \(\sqrt{7}\approx2.646\), we get \(-\sqrt{7}\approx-2.646\). As \(-2.646<-2.6\), A is to the left of B. Exam tip: among negative numbers, the value with greater magnitude lies further left.