Number line

Write the decimals with the same number of places: \(-0.11=-0.110\). Comparing \(-0.110\) and \(-0.101\), \(-0.110\) is more negative, so it lies farther left on the number line and is smaller. Therefore, \(-0.11\) is correct. \(-0.101\) is closer to zero, so it is greater. Exam tip: When comparing negative decimals, add trailing zeros if needed and then compare place values.

The midpoint of two numbers is found using \(\frac{a+b}{2}\). Here, \(\frac{-8+8}{2}=\frac{0}{2}=0\). Therefore, 0 is at an equal distance from -8 and 8. Option 4 is incorrect because it is not the average of -8 and 8. Exam tip: The midpoint of opposite numbers \(a\) and \(-a\) is always 0.

The distance between two numbers on a number line is the absolute value of their difference: \(|7-(-14)|=|21|=21\). Therefore, the correct answer is 21. Choosing 20 usually results from missing the sign change while subtracting a negative number. Exam tip: always use \(|a-b|\) to find distance.

Writing -0.2 as -0.20 only adds a zero to the right of the decimal part; it does not change the value of the number. Hence, -0.2 = -0.20, so both are equal. Options A and B are not correct because neither number is greater than the other. Exam tip: Zeros added at the end of a decimal do not change its value.

For negative numbers, the number closer to zero is greater. \(-0.00005\) is closer to zero than \(-0.0005\); therefore, \(-0.00005 > -0.0005\). They are not equal because their decimal values are different. Exam tip: On a number line, the number farther to the right is greater.

The midpoint of two numbers is their average: \(\frac{-13+9}{2}=\frac{-4}{2}=-2\). Hence, \(-2\) is correct. \(-1\) can result from an error while adding the signed numbers. Exam tip: add the numbers with their signs first, then divide by 2.

For negative numbers, the number farther to the right on the number line is greater. \(-20.5\) lies to the right of \(-21\) because it is closer to zero. Therefore, \(-20.5\) is greater. Although \(-20.75\) is also greater than \(-21\), it is not one of the two numbers being compared. Exam tip: among negative numbers, the number with the greater negative value is smaller.

For negative numbers, the number closer to zero is greater. \(-0.007\) is closer to zero than \(-0.07\), so \(-0.007 > -0.07\). Choosing \(-0.07\) is a common error because, among negative numbers, the one with the larger magnitude is smaller. Exam tip: write \(-0.07\) as \(-0.070\) before comparing decimal places.

Between any two distinct real numbers, there are infinitely many rational and infinitely many irrational numbers. Hence every non-zero interval on the number line contains both types. Exam tip: when you see “two distinct real numbers,” recall the density property.

The midpoint of two numbers on a number line is their average: \(\frac{-10+6}{2}=\frac{-4}{2}=-2\). Therefore, \(-2\) is correct. \(2\) may result from ignoring the negative sign, but the sum of \(-10\) and \(6\) is \(-4\). Exam tip: first add both numbers carefully, then divide the result by 2.

The distance between two numbers on a number line is the absolute value of their difference. Thus, \(|4-(-17)|=|21|=21\). Therefore, 21 is correct. Choosing 20 would result from missing the sign change when subtracting a negative number. Exam tip: always take the absolute value of the difference when finding distance.

The successor of an integer is obtained by adding 1. Thus, \(-31+1=-30\), so -30 is the correct answer. -32 is the predecessor of -31 because it lies one step to the left of -31 on the number line. Exam tip: moving one step right on a number line increases the number by 1.

The predecessor of a whole number is exactly 1 less than that number. Therefore, the predecessor of 20 is \(20-1=19\). Since 18 is 2 less than 20, it is not the predecessor. Exam tip: subtract 1 from the given number to find its predecessor.

The absolute value of a number is its distance from 0 on the number line. Since \(-19\) is 19 units away from 0, \(|-19|=19\). \(-19\) is the number itself, not its absolute value. Exam tip: the absolute value of a negative number is always positive.

Real numbers are dense on the number line. For example, between 1 and 2, \(3/2\) is rational while \(\sqrt{2}\) is irrational, and infinitely many more exist. Exam tip: unlike integers, real numbers have no consecutive pair.

The midpoint of two numbers on a number line is their average: \(\frac{-7+(-1)}{2}=\frac{-8}{2}=-4\). Therefore, \((-4)\) is the correct answer. \((-5)\) is closer to one of the numbers and is not equally distant from both. Exam tip: add the two numbers and divide by 2 to find their midpoint.

Write \(-2.5\) as \(-2.50\) to compare the decimals. Since \(-2.05\) is closer to zero and lies to the right of \(-2.50\) on the number line, \(-2.05\) is greater. Choosing \(-2.5\) is incorrect because, among negative numbers, the number with the larger magnitude is smaller. Exam tip: Add trailing zeros when needed before comparing decimals.

On the number line, the number to the right is greater.
( -0.0009 ) is a negative number and lies to the left of 0, so 0 is greater. “Both are equal” is incorrect because one number is zero and the other is negative. Exam tip: Zero is always greater than any negative number.

The distance between two numbers on a number line is the absolute value of their difference. Here, \(\lvert -5-(-12)\rvert=\lvert 7\rvert=7\), so 7 is correct. Getting 8 would result from an incorrect subtraction. Exam tip: a distance is always non-negative.

Between any two distinct rational numbers, there are infinitely many rational as well as irrational numbers. Hence C is correct; B is false because irrationals also occur between them. Exam tip: real numbers are dense on the number line.