Number line

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.

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.

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.

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.

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.

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.

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.

-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.

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.

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.

-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.

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.

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.

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.

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.

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.

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.

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.

-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.

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.