Number line

Correct answer: A. 3

Explanation: The midpoint of two numbers is their average: \(\frac{-3+9}{2}=\frac{6}{2}=3\). Therefore, the correct answer is 3. Note that 6 is the sum of the two numbers, not their midpoint. Exam tip: Add the two numbers and divide by 2 to find their midpoint on a number line.

Correct answer: B. \(-0.09\)

Explanation: For negative numbers, the number closer to zero is greater. Since \(-0.09\) is closer to zero than \(-0.9\), we have \(-0.09 > -0.9\). The options \(0.09\) and \(0.9\) are positive numbers, so they are not the numbers being compared. Exam tip: among negative decimals, the one with the smaller absolute value is greater.

Correct answer: B. -2.4

Explanation: -2.4 can be written as -2.40. Since -2.40 is more negative than -2.04, it lies farther to the left on the number line and is therefore smaller. Option -2.04 is negative, but it is greater than -2.4. Exam tip: Add trailing zeros when needed to compare decimal places easily.

Correct answer: C. \(2<P<3\)

Explanation: Moving to the right means adding the distance, so \(P=-1+\sqrt{13}\). As \(3.6<\sqrt{13}<3.7\), we get \(2.6<P<2.7\). Therefore, \(P\) lies between 2 and 3; do not treat the starting coordinate as zero.

Correct answer: A. 7

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-4+18}{2}=\frac{14}{2}=7\). Therefore, the correct answer is 7. A value such as 6 may result from an incorrect calculation of the sum or division. Exam tip: Include the signs of both numbers, add them, and then divide by 2.

Correct answer: A. 3

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-1+7}{2}=\frac{6}{2}=3\). Therefore, 3 is correct. Option 2 is not correct because it is not equally distant from -1 and 7. Exam tip: Add the two numbers and divide by 2 to find the midpoint.

Correct answer: B. -3.5

Explanation: -3.5 can be written as -3.50. On the number line, the more negative number lies farther to the left and is smaller. Therefore, -3.50 is smaller than -3.05. The options 3.05 and 3.5 are positive, so both are greater than either negative number. Exam tip: When comparing negative decimals, add zeros if needed to make the decimal places equal.

Correct answer: A. 3

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-9+15}{2}=\frac{6}{2}=3\). Hence, 3 is correct. \(-3\) is not at an equal distance from both numbers. Exam tip: add the two numbers and divide the result by 2 to find the midpoint.

Correct answer: B. -0.101

Explanation: The correct answer is -0.101. We can write -0.11 as -0.110. On comparing them, -0.101 is greater than -0.110 because, among negative numbers, the number closer to zero is greater. The two numbers are not equal. Exam tip: While comparing negative decimals, add trailing zeros to make the number of decimal places equal.

Correct answer: A. 10

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-2+22}{2}=\frac{20}{2}=10\). Therefore, 10 is correct. 11 is not correct because it is not equally distant from −2 and 22. Exam tip: add the two numbers and divide by 2 to find their midpoint.

Correct answer: A. 20

Explanation: The distance between two numbers on a number line is the absolute value of their difference: \(\lvert -40-(-20)\rvert=\lvert -20\rvert=20\). Therefore, the correct answer is 20. \(-20\) is the signed difference, not a distance, because distance cannot be negative. Exam tip: Always use the absolute value of the difference when finding distance.

Correct answer: B. \(-0.004\)

Explanation: \(-0.004\) is greater than \(-0.04\) because, among negative numbers, the number closer to zero is greater. On the number line, \(-0.004\) lies to the right of \(-0.04\). \(0.004\) and \(0.04\) are positive numbers, so they do not answer the comparison between the two given numbers. Exam tip: When comparing negative decimals, identify which number is closer to zero.

Correct answer: A. \( -2 \)

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-8+4}{2}=\frac{-4}{2}=-2\). Hence, \( -2 \) is correct. \( -4 \) is not equally distant from \( -8 \) and \( 4 \), so it is not the midpoint. Exam tip: add the two numbers and divide the result by 2 to find their midpoint.

Correct answer: B. -0.602

Explanation: Write the numbers to the same number of decimal places: -0.62 = -0.620. Now compare them: -0.602 is greater than -0.620 because, among negative numbers, the number closer to zero is greater. The positive numbers 0.602 and 0.620 are not valid answers for comparing the given negative numbers. Exam tip: Add trailing zeros when needed to make the decimal places equal before comparing decimals.

Correct answer: A. \(-15\)

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-11+(-19)}{2}=\frac{-30}{2}=-15\). Therefore, \(-15\) is correct. \(-14\) is not the midpoint because it is 3 units from \(-11\) and 5 units from \(-19\). Exam tip: add the numbers with their signs, then divide by 2.

Correct answer: A. 9

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-3+21}{2}=\frac{18}{2}=9\). Therefore, 9 is correct. Neither 8 nor 10 is at an equal distance from both numbers. Exam tip: Add the two numbers and divide the result by 2 to find their midpoint.

Correct answer: C. The statement is incorrect because infinitely many rational numbers lie between \(P\) and \(Q\).

Explanation: Since \(2<\left(\frac{3}{2}\right)^2=\frac94<3\), we get \(\sqrt2<\frac32<\sqrt3\). Thus \(\frac32\) is a rational number between them; in fact, infinitely many rationals lie between distinct real numbers. Exam tip: compare positive roots by squaring.

Correct answer: C. \(\sqrt{2}\)

Explanation: \(\sqrt{2}\) is irrational, and \(1.4^2=1.96<2<2.25=1.5^2\), so \(1.4<\sqrt{2}<1.5\). Thus, the interval contains an irrational number. Exam tip: compare squares of the endpoints to locate a square root.

Correct answer: A. 12

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-6+30}{2}=\frac{24}{2}=12\). Therefore, 12 is correct. For example, 18 is not at an equal distance from -6 and 30. Exam tip: add the two numbers and divide by 2 to find their midpoint.

Correct answer: B. \(-0.77\)

Explanation: Write \(-0.77\) as \(-0.770\) to compare the decimals. Since \(-0.770 < -0.707\), \(-0.77\) is smaller. The closest distractor, \(-0.707\), is closer to zero and is therefore greater. Exam tip: among negative decimals, the number farther left on the number line is smaller.

Correct answer: A. \(-15\)

Explanation: The midpoint of two numbers is their average: \(\frac{-9+(-21)}{2}=\frac{-30}{2}=-15\). Therefore, \(-15\) is correct. A value such as \(-14\) is not at an equal distance from both numbers. Exam tip: when adding negative numbers, add their magnitudes and retain the negative sign.

Correct answer: A. 8

Explanation: The midpoint of two numbers on a number line is their average: \(\frac{-2+18}{2}=\frac{16}{2}=8\). Therefore, 8 is correct. The value 9 is not the average of -2 and 18. 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: \(\lvert -35-(-15)\rvert=\lvert -20\rvert=20\). Therefore, the correct answer is 20. The option 18 comes from an incorrect subtraction. Exam tip: distance is always non-negative.

Correct answer: B. \(-0.509\)

Explanation: Write \(-0.51\) as \(-0.510\) to compare the decimals. Since \(-0.509\) is greater than \(-0.510\), \(-0.509\) is the correct answer. Among negative numbers, the number closer to zero is greater. Although \(-0.501\) would be greater than both given numbers, it is not one of the two numbers being compared. Exam tip: first make the number of decimal places equal before comparing decimals.

Correct answer: C. It contains infinitely many rational and infinitely many irrational numbers

Explanation: Between any \(a<b\), there are infinitely many rational as well as irrational numbers, so C is correct. In exams, remember that the real number line has no gaps between two real numbers.