Number line

The midpoint of two numbers on a number line is their average: \(\frac{-18+6}{2}=\frac{-12}{2}=-6\). Therefore, \( -6 \) is correct. Although \(0\) lies between the two numbers, it is not equally distant from both. Exam tip: add the two numbers and divide by 2 to find their midpoint.

\(\sqrt{2}\) cannot be expressed as \(\frac{p}{q}\), so it is irrational. \(-\frac{7}{4}\), \(0.125\), and \(0.\overline{3}\) are rational. Exam tip: terminating or recurring decimals are rational.

For negative numbers, the number closer to zero is greater. Here, \(-9.99\) is closer to zero than \(-10\), so \(-9.99 > -10\). “Both are equal” is incorrect because the two decimal values are different. Exam tip: On a number line, the number farther to the right is greater.

The midpoint of two numbers on a number line is their average: \(\frac{11+(-5)}{2}=\frac{6}{2}=3\). Therefore, 3 is correct. The number 2 is not at an equal distance from 11 and -5. Exam tip: While finding a midpoint, add a negative number using brackets to avoid a sign error.

Write -1.75 as -1.750. On the number line, -1.750 lies to the left of -1.705, so it is smaller. Among negative numbers, the number with the greater magnitude is smaller. Option A is a close distractor, but -1.705 is less negative and therefore greater. Exam tip: Add trailing zeros to make the decimal places equal before comparing.

On a number line, the distance between two numbers is the absolute value of their difference: \(\lvert -24-(-8)\rvert=\lvert -16\rvert=16\). Therefore, the correct answer is 16. The value 24 is the magnitude of one number, not the distance between the two numbers. Exam tip: A distance is always non-negative.

The successor of an integer is 1 greater than that integer. Thus, \( -55+1=-54 \), so \( -54 \) is correct. \( -56 \) is 1 less than \( -55 \), so it is the predecessor. Exam tip: To find a successor, add 1 even for negative integers.

For negative numbers, the number closer to zero is greater. \(-0.00001\) is closer to zero than \(-0.0001\), so \(-0.00001 > -0.0001\). Option A is smaller because it has a greater negative magnitude. Exam tip: When comparing negative decimals, the negative number with the smaller magnitude is greater.

\(2.1\) can be written as \(2.10\). Comparing \(2.10\) and \(1.99\), their whole-number parts are 2 and 1 respectively; therefore, \(2.10>1.99\). Hence, \(2.1\) is greater. Although \(1.99\) is very close to 2, it is still less than 2. Exam tip: When comparing decimals, add trailing zeroes if needed to make the number of decimal places equal.

The midpoint of two numbers is their average: \(\frac{-16+0}{2}=\frac{-16}{2}=-8\). Thus, \(-8\) is equally distant from \(-16\) and \(0\). \(-7\) is not the midpoint because it is not at the same distance from both numbers. Exam tip: use \(\frac{a+b}{2}\) to find the midpoint on a number line.

Write the numbers to the same number of decimal places: \(-5.55=-5.550\). Comparing \(-5.555\) and \(-5.550\), \(-5.555\) is more negative, so it lies farther left on the number line and is smaller. Thus, \(-5.550\) is greater. Exam tip: among negative decimals, the number with the greater magnitude is the smaller number.

On a number line, the distance between two numbers is the absolute value of their difference: \(|-100-0|=|-100|=100\). Therefore, the correct answer is 100. The value 99 would be the distance from -100 to -1, not from -100 to 0. Exam tip: distance is always non-negative.

Both sets are dense. Choose rational \(r\) between \(a\) and \(b\); for large \(n\), \(r+\sqrt{2}/n\) stays in the interval and is irrational. Thus C is correct. Exam tip: dense means infinitely many numbers in every interval.

The distance between two numbers on a number line is the absolute value of their difference: \(\lvert -28-(-14)\rvert=\lvert -14\rvert=14\). Therefore, the correct answer is 14. Values such as 13 and 15 are one less or one more, so they do not represent the distance. Exam tip: a distance is always non-negative.

The correct answer is -0.49. On a number line, the negative number closer to zero is greater. Since -0.49 is closer to zero than -0.5, we have -0.49 > -0.5. “Both are equal” is incorrect because the decimal values are different. Exam tip: among negative numbers, the number with the smaller magnitude is greater.

The midpoint of two numbers on a number line is their average: \(\frac{-7+13}{2}=\frac{6}{2}=3\). Therefore, 3 is the correct answer. The number 2 is not the midpoint because it is not equally distant from -7 and 13. Exam tip: add the two numbers and divide by 2 to find their midpoint.

-13.5 can be written as -13.50. On a number line, the more negative number lies farther to the left and is smaller. Since -13.50 < -13.05, -13.5 is the smaller number. Although -13.05 looks similar, it is closer to zero and is therefore greater. Exam tip: write negative decimals with the same number of decimal places before comparing them.

The midpoint of two numbers is their average: \(\frac{-9+(-2)}{2}=\frac{-11}{2}=-5.5\). Therefore, \(\displaystyle -\frac{11}{2}\) is correct. Although \(-6\) lies between the two numbers, it is not at equal distance from both. Exam tip: add the two coordinates and divide by 2 to find a midpoint on the number line.

The distance between two numbers on a number line is the absolute value of their difference. Thus, \(\left|-3-(-15)\right|=\left|12\right|=12\). Therefore, \(12\) is correct. \(18\) is obtained by adding the absolute values of the numbers, but distance must be found using the absolute difference. Exam tip: distance can never be negative.

-0.11 can be written as -0.110. On comparing, -0.101 is closer to zero than -0.110, so -0.101 is greater. The numbers are not equal because their decimal digits differ. Exam tip: Among negative numbers, the number closer to zero is greater.