Number line

The midpoint of two numbers on a number line is their average: \(\frac{-6+6}{2}=\frac{0}{2}=0\). Therefore, \(0\) is at an equal distance from \(-6\) and \(6\). The number \(3\) is the midpoint of \(0\) and \(6\), not of \(-6\) and \(6\). Exam tip: The midpoint of opposite numbers \(a\) and \(-a\) is always \(0\).

On a number line, the distance between two numbers is the absolute value of their difference: \(\lvert -11-(-3)\rvert=\lvert -8\rvert=8\). Therefore, the correct answer is 8. Choosing 7 may result from undercounting, whereas distance is never negative. Exam tip: always take the absolute value of the difference when finding distance.

( -2.2 ) can be written as ( -2.20 ). On the number line, ( -2.22 ) lies to the left of ( -2.20 ), so it is more negative and therefore smaller. ( -2.2 ) is the closest distractor, but it is greater than ( -2.22 ). Exam tip: when comparing negative decimals, the more negative number is the smaller one.

\(2\) is greater than \(-3\) and less than \(8\), so \(-3 < 2 < 8\). Therefore, \(2\) is an integer between the two numbers. \(-4\) and \(-6\) are less than \(-3\), while \(9\) is greater than \(8\). Exam tip: For “between” questions, check that the number is greater than the left endpoint and less than the right endpoint.

The midpoint of two numbers is their average: \(\frac{-10+5}{2}=\frac{-5}{2}=-2.5\). Hence, \( -2.5 \) is correct. \(2.5\) results from a sign error, since the sum is \(-5\). Exam tip: When averaging a negative and a positive number, check the sign of their sum carefully.

The distance between two numbers on a number line is the absolute value of their difference: \(|2-(-14)|=|16|=16\). Therefore, the correct answer is 16. A distance cannot be negative, so -16 is incorrect; 12 would correspond to a different pair of numbers. Exam tip: Always use absolute value when finding distance on a number line.

The correct answer is
-0.003. Among negative numbers, the number closer to zero is greater. Since -0.003 is closer to zero than -0.03,
-0.003 > -0.03. The close distractor -0.03 is smaller because it has a greater negative magnitude. Exam tip: Write negative decimals to the same number of decimal places, as -0.003 and -0.030, before comparing them.

Since \(9<10<16\), we get \(3<\sqrt{10}<4\). On multiplying by \(-1\), the order reverses, so \(-4<-\sqrt{10}<-3\). Option B wrongly places \(\sqrt{10}\) below 3. Exam tip: locate square roots using nearby perfect squares.

The midpoint of two numbers on a number line is their average: \(\frac{-7+7}{2}=\frac{0}{2}=0\). Therefore, \(0\) is equally distant from \(-7\) and \(7\). Option \(1\) is not the midpoint because it is not at the same distance from both numbers. Exam tip: Add the two numbers and divide by 2 to find their midpoint.

The distance between two numbers on a number line is their absolute difference: \(\lvert -9-(-2)\rvert=\lvert -7\rvert=7\). Hence, the correct answer is 7. Choosing 6 results from an incorrect subtraction. Exam tip: a distance is always non-negative.

Write \(-1.01\) as \(-1.010\) to compare the decimals. On the number line, \(-1.010\) lies to the left of \(-1.001\), so \(-1.01\) is smaller. \(-1.001\) is closer to zero, so it is greater. Exam tip: among negative decimals, the number farther from zero is the smaller number.

The midpoint of two numbers on a number line is their average: \(\frac{2+(-5)}{2}=\frac{-3}{2}=-1.5\). Hence, \(-1.5\) is correct. \(1.5\) may result from incorrectly ignoring the negative sign. Exam tip: write negative numbers in brackets while substituting in the average formula.

On a number line, the distance between two numbers is the absolute value of their difference. Thus, \(\lvert 6-(-12)\rvert=\lvert 18\rvert=18\). Therefore, 18 is correct. The value 12 is only the absolute value of -12, not the distance between the two numbers. Exam tip: always take the absolute value of the difference when finding distance.

Infinitely many rational and irrational numbers lie between two distinct rational numbers, so C is correct. “Only rationals” is incomplete. Exam tip: treat “between” as an open interval.

The zero after the decimal does not change the value of \(-15.0\). Thus, \(-15.0=-15\), and both represent the same point on the number line. Choosing either \(-15\) or \(-15.0\) as greater is incorrect because they are equal. Exam tip: Zeros written at the end of a decimal do not change its value.

The midpoint of two numbers on a number line is their average: \(\frac{-4+9}{2}=\frac{5}{2}=2.5\). Therefore, 2.5 is correct. Although 3 is close to 2.5, it is not equally distant from both numbers. Exam tip: add the two numbers and divide by 2 to find their midpoint.

For negative numbers, the number closer to zero is greater. \( -18 \) is closer to zero than \( -18.25 \); on a number line, \( -18 \) lies to the right of \( -18.25 \). Therefore, \( -18 \) is greater. \( -18.5 \) is smaller because it is more negative. Exam tip: when comparing negative numbers, the less negative number is greater.

Rational numbers are dense on the number line, so a rational number lies between any two distinct real numbers, including irrationals. Since \(2<\frac{9}{4}<3\), \(\sqrt{2}<\frac{3}{2}<\sqrt{3}\). Exam tip: test “always” statements using a counterexample.

Both rational and irrational numbers lie between any two distinct real numbers, so C is correct. There is no immediate next rational number after a rational number. Exam tip: remember that the real number line is dense.

The midpoint of two numbers on a number line is their average: \(\frac{-9+11}{2}=\frac{2}{2}=1\). Therefore, the correct answer is \(1\). The value \(-1\) may result from an incorrect calculation of the average. Exam tip: always use \(\frac{x+y}{2}\) to find the midpoint of two numbers.