Representing real numbers on the number line

(0.999\ldots=1), so both are the same point. Do not treat a recurring decimal like a terminating decimal.

Since (2^2=4) and (3^2=9), (\sqrt{7}) lies between (2) and (3). Bound the number using nearby squares.

(\sqrt{8}) lies between (2) and (3), so (-\sqrt{8}) lies between (-3) and (-2). The negative sign changes the direction.

The number 2.10 can also be written as 2.1. Comparing the decimals by place value, the tenths digit of 2.10 is 1, while that of 2.01 is 0. Hence, 2.10 is greater and lies farther to the right on the number line. Exam tip: Compare the whole-number part first, followed by the tenths, hundredths, and later places.

(\frac{2}{5}=0.4), (\frac{1}{2}=0.5), and (\frac{3}{5}=0.6), so (\frac{1}{2}) lies between them. Decimal form helps in comparison.

\(\frac{-6}{2}=-3\) because 6 divided by 2 is 3, and the result remains negative since the numerator is negative. Therefore, the number is represented by the point -3 on the number line. Option 3 ignores the negative sign. Exam tip: simplify the fraction and check its sign before locating the point.

A fraction represents division of its numerator by its denominator. Therefore 8/4 means 8 divided by 4, which equals 2 because four fits into eight exactly two times. The fraction can also be simplified by dividing numerator and denominator by their common factor 4: 8/4 = (8÷4)/(4÷4) = 2/1 = 2. Hence the point 8/4 coincides with the integer 2 on the number line, so option A is correct. Option B confuses the denominator with the value of the fraction; option C repeats the numerator rather than performing the division; and option D would represent 1, which is obtained only when numerator and denominator are equal. The governing concept is fraction simplification and the identification of equivalent numerical values.

(x\le 2) means (2) or smaller numbers, so it is the left side including (2). The symbol (\le) includes the boundary point.

(x>-1) means numbers greater than (-1), so the region is to the right and (-1) is not included. The symbol (>) gives an open point.

The number exactly midway between two numbers is their average. Thus, \(\frac{1+4}{2}=\frac{5}{2}=2.5\). Hence, 2.5 is correct because it is 1.5 units from both 1 and 4. The number 3 is closer to 4, while 2 is closer to 1. Exam tip: Find the midpoint of \(a\) and \(b\) using \(\frac{a+b}{2}\).

Since (3^2=9) and (4^2=16), (\sqrt{13}) lies between (3) and (4). Use nearby perfect squares to locate square roots.

(-\frac{9}{4}=-2.25), so it lies between (-3) and (-2). Be careful with direction for negative fractions.

The exact midpoint of two numbers is their arithmetic mean, found by adding the endpoints and dividing by 2. For −2 and 3, the calculation is (−2 + 3)/2 = 1/2. Thus 1/2 is the number equally distant from both endpoints: the distance from −2 to 1/2 is 2.5 units, and the distance from 1/2 to 3 is also 2.5 units. Therefore option A is correct. Option B is one unit from 0 but is not equally distant from the given endpoints; its distances are 3 and 2. Option C lies on the negative side and is not the average, while option D is outside the interval altogether. The governing concept is the midpoint formula (a+b)/2, which works for positive, negative, and mixed endpoints alike.

(\frac{3}{4}) is greater than (0) and less than (1). In exams first compare the fraction value with nearby integers.

The governing concept is the order of integers on a number line. A number lies between two consecutive integers when it is greater than the left integer and less than the right integer. For −2.5, the correct comparison is −3 < −2.5 < −2. Thus −2.5 lies between −3 and −2, exactly halfway between them. Negative numbers require careful attention because the integer with the larger absolute value is farther left: −3 is left of −2. Option B describes the interval immediately to the right of −2, option C contains positive numbers, and option D is the interval to the left of −3. Therefore, option A is the only interval containing −2.5.

The principal square root of a number is the non-negative number that gives the number when multiplied by itself. Therefore, \(\sqrt{9}\) means the non-negative number whose square is 9. On a number line, this value is located three units to the right of zero, so it corresponds to point 3. The correct choice is B.

To verify the result, calculate the squares of the relevant choices: \(2^2=4\), \(3^2=9\), \(4^2=16\), and \(9^2=81\). Only 3 has square 9. Although \(-3\) also satisfies \((-3)^2=9\), the symbol \(\sqrt{9}\) refers specifically to the principal, non-negative root, not both solutions. Thus the answer is 3.

Since (2^2<5<3^2), (\sqrt{5}) lies between (2) and (3). Use squares to locate square roots quickly.

Moving right from (0) gives a positive number, so (A=1.2). Check direction first to decide the sign.

(-\frac{1}{2}=-0.5), so both represent the same point. Converting fractions to decimals makes comparison easier.

(1.75=1+\frac{3}{4}), so it lies at the third equal part after (1). Converting decimals to fractions helps locate points easily.