Representing real numbers on the number line

(\frac{11}{5}=2.2), so it lies between (2) and (3). Convert an improper fraction to decimal or mixed form.

Since (2^2<7<3^2), (\sqrt{7}) lies between (2) and (3). Taking (\sqrt{7}) as (7) is a common mistake.

The governing concept is distance on a number line: the distance between a number x and 3 is |x − 3|. For 3.04, the distance is |3.04 − 3| = 0.04. For 3.4, it is |3.4 − 3| = 0.4. Because 0.04 is smaller than 0.4, the point 3.04 lies closer to 3. Equivalently, 3.04 is only four hundredths to the right of 3, whereas 3.4 is four tenths to the right. Therefore option A is correct. Option B confuses the larger decimal displacement with greater closeness, while option C is false because the distances are not equal. Option D is also false because both points have definite positions.

(-0.3) is greater than (-0.8), so it is to the right. Among negative numbers, the one closer to (0) is greater.

For (\sqrt{n}) to lie between (4) and (5), (16<n<25) is needed. Among the options (21) is correct.

The midpoint of two numbers is their average. Thus, \(\frac{1.2+1.3}{2}=\frac{2.5}{2}=1.25\). Therefore, the correct answer is 1.25. Although 1.21 is close to 1.2, it is not equidistant from 1.2 and 1.3. Exam tip: Add the two endpoint values and divide the sum by 2 to find their midpoint.

The governing number-line rule is that the greater real number lies farther to the right. Convert the fraction to a decimal: 3/2 = 1.5. Comparing 1.5 and 1.6 gives 1.6 > 1.5, so 1.6 has the greater value and must be located to the right of 3/2. Hence option B is correct. Option A would be correct only if 1.5 were greater than 1.6, which it is not. Option C is incorrect because the values differ, and option D is incorrect because 1.6 is greater than 1, while 3/2 is also 1.5, so neither number is less than 1. Converting both representations to a common form avoids confusion.

Dividing 9 by 5 gives 1.8, and the negative sign makes the value \(-1.8\). Thus, it is located 1.8 units to the left of zero on the number line. Option B has an incorrect decimal-place value, while option C omits the negative sign. Exam tip: After converting a fraction to a decimal, always check whether its sign has been retained.

(\sqrt{18}) is about (4.24), so it is closer to (4). For estimation, check nearby perfect squares (16) and (25).

The governing concept is directed movement on a number line. Moving to the right increases a coordinate, while moving to the left decreases it. Starting at 2 and moving 1.5 units left therefore requires subtraction: Q = 2 − 1.5 = 0.5. Thus option B is correct. A quick check is that 0.5 is indeed 1.5 units to the left of 2, because 2 − 0.5 = 1.5. Option A comes from adding instead of subtracting, option C gives the distance moved rather than the final coordinate, and option D would be obtained by moving 2.5 units left or by using an incorrect subtraction. The direction word “left” determines the operation.

The value of \(\sqrt{2}\) is approximately \(1.414\). Therefore, \(1+\sqrt{2}\approx 1+1.414=2.414\), which lies between 2 and 3. Hence, option B is correct. Exam tip: For such questions, estimate the irrational number first and then identify the consecutive integers containing the result.

Converting \(\frac{3}{4}\) into decimal form gives \(3 \div 4=0.75\). Therefore, \(\frac{3}{4}\) and 0.75 represent the same point on the number line. The other pairs are unequal; for example, \(\frac{2}{5}=0.4\), not 0.25. Exam tip: convert the fraction into decimal form and compare the values.

(-\sqrt{9}=-3) and (\sqrt{9}=3), so the distance is (6). The distance between opposite points is the sum of their magnitudes.

Write the decimals with equal place values: 0.09, 0.19, and 0.90. Comparing the hundredths and tenths places gives 9 < 19 < 90, so the increasing order is 0.09 < 0.19 < 0.9. Option D is incorrect because 0.9 is greater than 0.19, not smaller. Exam tip: Add trailing zeroes when needed so that decimals have the same number of places before comparing them.

We are given \(5<\sqrt{a}<6\). Since all quantities are positive, squaring all three parts preserves the inequality and gives \(25<a<36\). Option A incorrectly treats the range of \(\sqrt{a}\) as the range of \(a\). Exam tip: for positive bounds involving a square root, square both boundary values.

Since \(\sqrt{2}\approx 1.414\), we get \(2-\sqrt{2}\approx 2-1.414=0.586\). This value lies between \(0\) and \(1\). Option C is incorrect because \(2-\sqrt{2}\) is less than \(1\). Exam tip: estimate the irrational number first, perform the operation, and then identify the consecutive integers on either side of the result.

(\frac{-13}{6}\approx -2.17), so it lies between (-3) and (-2). Intervals with negative numbers can be tricky.

On a number line, the length between two points is the distance separating their coordinates. Distance cannot be negative, so we use the absolute value of the difference. If one point is at 1.4 and the other is at 2.1, the second point lies to the right of the first. The segment length is therefore found by subtracting the smaller coordinate from the larger one.

Using the distance rule, write AB = \(|2.1 - 1.4|\). The subtraction gives 0.7, and its absolute value is also 0.7. Thus the points are seven-tenths of one unit apart, so option B is correct. The value 1.7 would come from adding the coordinates, which does not measure the distance between points on a number line.

Moving right means adding 2.25, while moving left means subtracting 0.5. Thus, the final point is \(0+2.25-0.5=1.75\). Option A can result from an incorrect subtraction; remember to add for rightward movement and subtract for leftward movement.

(\sqrt{3}) is about (1.732), so (1.7) is closest. For estimation place (\sqrt{3}) between (1) and (2).