Representing real numbers on the number line

Correct answer: A. (3)

Positive numbers lie to the right of (0) on the number line. The value increases as we move right.

Correct answer: A. -4

Negative numbers are located to the left of 0 on the number line, so -4 is correct. The numbers 1, 2, and 5 are positive and lie to the right of 0. Exam tip: moving left on the number line decreases the value of a number.

Correct answer: A. 2.5

The governing idea is the order relation of real numbers on a number line. A number x lies strictly between 2 and 3 when it satisfies both inequalities 2<x and x<3, or together 2<x<3. Test option A: 2<2.5<3, so 2.5 is located to the right of 2 and to the left of 3. Option B, 3.5, is greater than 3. Option C, 1.5, is less than 2, and option D, 4.5, is also greater than 3. Therefore only option A lies in the required interval. The endpoints 2 and 3 would not count under the usual meaning of “between,” because the question asks for an interior point. A decimal such as 2.1 or 2.9 would also work, but among the given choices 2.5 is the only valid answer.

Correct answer: A. (0) and (1)

(\frac{1}{2}=0.5), so it lies between (0) and (1). Thinking of a fraction as a decimal helps locate it.

Correct answer: A. 2 units

The distance between two numbers is the absolute value of their difference: \(|6-4|=2\) units. Therefore, option A is correct. The 10-unit distractor may result from adding the numbers, but distance is found by taking their difference. Exam tip: Distance on a number line is always positive.

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

\(\sqrt{2}\) is irrational because it cannot be written as \(\frac{p}{q}\), and its decimal expansion is non-terminating and non-repeating. In contrast, 0.375 is a terminating decimal, so it is rational. Exam tip: the square root of a non-perfect square is usually irrational.

Correct answer: A. -(2) / (-2)

The greater number lies to the right and (-2>-5). Among negative numbers, the one closer to (0) is greater.

Correct answer: A. \\(\\frac{5}{4}\\)

The decimal number \\(1.25=\\frac{125}{100}=\\frac{5}{4}\\). Therefore, option A is correct. Option B, \\(\\frac{4}{5}\\), is equal to \\(0.8\\), not \\(1.25\\). Exam tip: When a decimal has two digits after the decimal point, write it over 100 and then reduce the fraction.

Correct answer: A. (4) parts

The denominator of (\frac{3}{4}) is (4), so divide (0) to (1) into (4) equal parts. Then move (3) parts to the right.

Correct answer: A. (1) and (2)

(\frac{7}{4}=1.75), so it lies between (1) and (2). Think of an improper fraction in mixed or decimal form.

Correct answer: A. \(\frac{1}{5}\)

Since 0.2 has one digit after the decimal point, it can be written as \(\frac{2}{10}\). Dividing the numerator and denominator by 2 gives \(\frac{2}{10}=\frac{1}{5}\), so option A is correct. In exams, count the decimal places to choose the denominator as 10, 100, and so on.

Correct answer: A. \(\frac{3}{4}\)

Converting the decimal to a fraction gives \(0.75=\frac{75}{100}=\frac{3}{4}\). Therefore, the point lies between 0 and 1, at a distance of \(\frac{3}{4}\) from 0. Option B, \(\frac{1}{4}\), represents 0.25, not 0.75. Exam tip: convert a terminating decimal into a fraction and reduce it to the simplest form.

Correct answer: A. 1 and 2

Since \(1^2=1<2<4=2^2\), taking positive square roots gives \(1<\sqrt{2}<2\). Therefore, \(\sqrt{2}\) lies between 1 and 2 on the number line. Exam tip: locate a square root by comparing the number with consecutive perfect squares; hence 0 and 1 is not correct.

Correct answer: A. (1) and (2)

Since (1^2<3<2^2), (\sqrt{3}) lies between (1) and (2). For square roots, check nearby perfect squares.

Correct answer: A. (2) and (3)

Since (2^2<5<3^2), (\sqrt{5}) lies between (2) and (3). Perfect squares quickly give the interval.

Correct answer: A. \(3\)

Since \(9=3^2\), \(\sqrt{9}=3\), so it represents the point 3 on the number line. Although \((-3)^2=9\), the square-root symbol denotes the principal, non-negative root. Exam tip: remember that \(\sqrt{a^2}=|a|\), not always \(a\).

Correct answer: A. 4

The governing concept is the principal square root. The symbol √16 represents the nonnegative number whose square is 16. Since 4^2=4×4=16, √16=4, so the corresponding point on the number line is 4 and option A is correct. A common distinction is needed here: the equation x^2=16 has two solutions, x=4 and x=-4, but the radical symbol √16 denotes only the principal, nonnegative square root. Thus -4 is not the value of √16. Option B is impossible because 8^2=64, and option C is incorrect because 16^2=256. The number 4 is therefore the unique value represented by the radical and the unique correct point among the choices.

Correct answer: A. (3) and (4)

Since (3^2<10<4^2), (\sqrt{10}) lies between (3) and (4). Always place (n) between nearby perfect squares.

Correct answer: A. \(|a-b|\)

The distance between \(a\) and \(b\) on the number line is \(|a-b|\). The absolute value ensures that the distance is never negative. Using only \(a-b\) may give a negative result, so remember to use the absolute value when finding distance.

Correct answer: A. 3

The midpoint of two numbers is their average. Therefore, the midpoint is \(\frac{1+5}{2}=\frac{6}{2}=3\). The value 4 is not correct because it does not divide the interval from 1 to 5 into two equal parts. In an exam, use the formula \(\frac{a+b}{2}\) to find the midpoint.

Correct answer: A. -1

The midpoint is the average of the two coordinates: (-4 + 2) ÷ 2 = -2 ÷ 2 = -1. Therefore, the midpoint’s coordinate is -1.

Correct answer: A. 5

Moving to the right on a number line means adding units. Therefore, moving 5 units right from 0 gives 0 + 5 = 5. The point -5 would be reached by moving 5 units to the left. Exam tip: rightward movement represents positive change, while leftward movement represents negative change.

Correct answer: A. −6

Moving 6 units to the left from 0 decreases the value by 6, so the point reached is 0 − 6 = −6. Option B, 6, would be reached by moving 6 units to the right. Exam tip: on a number line, the left direction represents negative movement and the right direction represents positive movement.

Correct answer: A. 5

On a number line, moving to the right means adding the distance travelled. Therefore, moving 3 units right from 2 gives 2 + 3 = 5. The point -1 would be reached by moving 3 units left, while 3 is only the distance travelled, not the final point. Exam tip: add when moving right and subtract when moving left.

Correct answer: A. -3

Moving left on the number line decreases the value. Therefore, moving 5 units left from 2 gives 2 - 5 = -3. Option B results from adding instead of subtracting. Exam tip: subtract for a leftward movement and add for a rightward movement.