Representing real numbers on the number line

Correct answer: B. 4 and 5

Since \(6^2=36<40<49=7^2\), we get \(6<\sqrt{40}<7\). Subtracting 2 from all parts gives \(4<\sqrt{40}-2<5\), so the number lies between 4 and 5. Option A is incorrect because the value is greater than 4. Exam tip: bound the square root using consecutive perfect squares, then apply the operation to the entire inequality.

Correct answer: B. 7.5

The distance between two points on a number line is the absolute difference of their coordinates: \(|AB|=|2.9-(-4.6)|=|7.5|=7.5\). Therefore, the correct answer is 7.5. The distractor 6.5 may result from mishandling the subtraction of a negative number. Exam tip: Always take the absolute value when finding distance, since distance cannot be negative.

Correct answer: B. The (11)th point among (12) equal parts from (0) to (1)

The direct answer is option B: the 11th point after dividing the interval from 0 to 1 into 12 equal parts. A fraction \(\frac{11}{12}\) means 11 equal parts out of a total of 12. Divide the unit interval into 12 equal pieces; each piece has length \(\frac{1}{12}\). Starting at 0 and taking 11 such pieces reaches \(\frac{11}{12}\), just before 1. Option A incorrectly says the 12th point among 11 parts; it reverses the numerator and denominator. Option B correctly uses 12 as the number of equal divisions and 11 as the number of parts taken. Option C, 11 units to the right of 1, represents 12, not \(\frac{11}{12}\). Option D, 12 units left of 0, represents -12. The denominator tells how many equal parts make one whole, while the numerator tells how many parts are selected. Memory cue: denominator divides, numerator counts.

Correct answer: A. ( \frac{1}{2} ) and ( \frac{15}{2} )

( |x-4|=\frac{7}{2} ) means (x) is at distance ( \frac{7}{2} ) from (4). Moving both directions gives ( \frac{1}{2} ) and ( \frac{15}{2} ).

Correct answer: A. (3\sqrt{2})

The direct answer is option A: \(3\sqrt2\). Simplify the perfect-square factor inside \(\sqrt8\): since \(8=4\times2\), \(\sqrt8=\sqrt4\sqrt2=2\sqrt2\). Thus \(\sqrt2+\sqrt8=\sqrt2+2\sqrt2=3\sqrt2\). These are like radicals because both contain \(\sqrt2\), so their coefficients 1 and 2 can be added. Option A is correct. Option B, \(\sqrt{10}\), incorrectly combines the numbers under separate square roots; in general \(\sqrt a+\sqrt b\ne\sqrt{a+b}\). Option C, \(2\sqrt{10}\), makes the same invalid combination and adds an extra factor. Option D, \(4\sqrt2\), treats \(\sqrt8\) as \(3\sqrt2\) instead of \(2\sqrt2\). A useful check is that \(\sqrt2\approx1.414\), so the sum is about 4.242, matching \(3\sqrt2\). Memory cue: simplify each radical first, then combine only like radicals.

Correct answer: A. ( \frac{1}{3} )

The direct answer is option A: \(1/3\). The midpoint of two numbers is their average, so add the endpoints and divide by 2: \(M=\frac{(-5/3)+(7/3)}2\). Since the denominators are already equal, combine the numerators: \(-5/3+7/3=2/3\). Then divide by 2: \((2/3)/2=(2/3)(1/2)=1/3\). Therefore option A is correct. Option B, \(2/3\), stops after adding the endpoints and forgets to divide by 2. Option C, 1, is not the average; it does not follow from the required calculation. Option D, \(-1/3\), has the wrong sign because the positive endpoint is larger in magnitude than the negative endpoint, so the midpoint must lie to the right of zero. As a check, \(1/3\) is equally distant from both endpoints. Memory cue: midpoint equals first point plus second point, all divided by 2.

Correct answer: B. ( \frac{9}{4} )

( \sqrt{5}\approx2.236 ) and ( \frac{9}{4}=2.25 ), so ( \frac{9}{4} ) is greater. The greater number lies farther right.

Correct answer: C. 91

We have \\(9<\sqrt{n}<10\\). Since both bounds are positive, squaring gives \\(81<n<100\\). The value 91 lies in this interval, so it is possible. The closest distractor is 100, but it is an endpoint and does not satisfy the condition that \\(\sqrt{n}<10\\). Exam tip: when squaring an inequality whose quantities are positive, the inequality signs remain unchanged.

Correct answer: B. ( \frac{1}{2} )

( \frac{4}{9}\approx0.444 ), ( \frac{1}{2}=0.5 ), and ( \frac{5}{9}\approx0.556 ). Hence ( \frac{1}{2} ) lies between them.

Correct answer: A. (p>q)

The important idea is that numbers farther to the right on a number line are greater. Both given numbers are negative, so their positions must be compared carefully: a negative number closer to zero is greater than a negative number farther from zero.

Since
\(\sqrt{17}\approx4.123\), we get \(p=-\sqrt{17}\approx-4.123\), while \(q=-4.2\). On the number line, \(-4.123\) lies to the right of \(-4.2\), so \(-4.123>-4.2\). Therefore, statement A, \(p>q\), is correct. Statement C reverses this comparison, and the other choices do not describe the given numbers.

Correct answer: B. ( -\frac{1}{3} )

The direct answer is option B, \( -\frac{1}{3} \). On a number line, the distance between two points is the absolute difference of their coordinates. If a point must lie to the left of \(\frac{3}{2}\), we move left, so we subtract the distance: \(\frac{3}{2}-\frac{11}{6}\). Convert \(\frac{3}{2}\) to sixths: \(\frac{3}{2}=\frac{9}{6}\). Therefore \(\frac{9}{6}-\frac{11}{6}=-\frac{2}{6}=-\frac{1}{3}\). Checking confirms that \(\left| -\frac{1}{3}-\frac{3}{2}\right|=\left| -\frac{2}{6}-\frac{9}{6}\right|=\frac{11}{6}\), and \(-\frac{1}{3}\) is to the left. Option A, \(\frac{10}{3}\), is to the right and its distance is not \(\frac{11}{6}\). Option B is correct because it satisfies both distance and direction. Option C, \(\frac{1}{6}\), is also to the left, but its distance from \(\frac{3}{2}\) is \(\frac{4}{3}\), not \(\frac{11}{6}\). Option D, \(-\frac{11}{6}\), is to the left, but its distance is \(\frac{10}{3}\). Memory cue: left means subtract; right means add, and always check the absolute distance afterward.

Correct answer: C. (2.499) and (2.501)

Both (2.499) and (2.501) are at distance (0.001) from (2.5). Points at equal distance are equally close.

Correct answer: C. ( \sqrt{5} )

By Pythagoras, the hypotenuse is ( \sqrt{2^2+1^2}=\sqrt{5} ). Right triangles help in square-root construction.

Correct answer: B. Irrational number

This decimal is non-terminating and non-repeating, so it is irrational. Every irrational number also has a place on the number line.

Correct answer: A. ( \sqrt{7}<2.65 )

( \sqrt{7}\approx2.646 ), so it is slightly less than (2.65). Compare close values to more decimal places.

Correct answer: B. 0.66

Since \(\sqrt{75}=5\sqrt{3}\approx5\times1.732=8.66\), we get \(a-8\approx8.66-8=0.66\). Therefore, option B is correct. Values such as 0.36 or 0.06 may result from errors in estimating the square root or subtracting the integer. Exam tip: estimate the square root first and then perform the subtraction.

Correct answer: C. ( -2.43 )

( -\sqrt{6}\approx-2.449 ), so (-2.43) lies between it and (-2.4). Read negative intervals carefully.

Correct answer: C. ( \frac{355}{113}>\pi )

( \frac{355}{113}\approx3.14159292 ) and ( \pi\approx3.14159265 ), so the fraction is slightly larger. Do not treat approximations as exactly equal.

Correct answer: B. (5.4<\sqrt{30}<5.5)

(5.4^2=29.16) and (5.5^2=30.25), so ( \sqrt{30} ) lies between them. Check squares for decimal bounds.

Correct answer: C. Between −1 and 0

Since \(4^2<22<5^2\), we get \(4<\sqrt{22}<5\). Adding \(-5\) to all parts gives \(-1<-5+\sqrt{22}<0\). Therefore, \(x\) lies between \(-1\) and \(0\). Option B is incorrect because \(x\) is negative. Exam tip: To bound a square-root expression, first compare the radicand with the squares of consecutive integers.

Correct answer: A. \(\frac{3}{8}\)

Since 0.375 has three decimal places, \(0.375=\frac{375}{1000}\). Dividing the numerator and denominator by 125 gives \(\frac{375}{1000}=\frac{3}{8}\), so option A is correct. The nearby distractor \(\frac{5}{8}\) equals 0.625, not 0.375. Exam tip: write a terminating decimal over \(10^n\), then reduce the fraction by their greatest common factor.

Correct answer: A. ( -\frac{11}{20} )

Moving right gives ( -\frac{5}{4}+\frac{7}{10}=-\frac{11}{20} ). Use a common denominator to add fractions.

Correct answer: C. (10)

The direct answer is option C: 10. To locate \(\sqrt{98}\), compare 98 with nearby perfect squares: \(9^2=81\) and \(10^2=100\). Hence \(\sqrt{98}\) lies between 9 and 10. Because 98 is very close to 100, its square root is very close to 10. Numerically, \(\sqrt{98}\approx9.899\). Its distance from 10 is about 0.101, while its distance from 9 is about 0.899, so 10 is the nearest integer. Option C is correct. Option A, 8, is too far away; \(8^2=64\). Option B, 9, is the lower bounding integer but is not nearest. Option D, 11, is above 10 and clearly farther; \(11^2=121\). A common error is to choose the lower integer merely because the root is below 10. Memory cue: bracket the root with consecutive squares, then compare the distances or note which square it is closer to.

Correct answer: C. 5

We have \(-3< -\sqrt{m}< -2\). Multiplying all parts by \(-1\) reverses the inequality signs, giving \(2<\sqrt{m}<3\). Since both sides are positive, squaring yields \(4<m<9\). Among the given choices, only \(5\) lies in this interval. For instance, \(3\) gives \(\sqrt{m}<2\), while \(10\) gives \(\sqrt{m}>3\), so they are not possible. Exam tip: reverse the inequality sign whenever you multiply or divide by a negative number.

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

The direct answer is option A: between \(-1\) and 0. Estimate the root using nearby squares: \(4^2=16\) and \(5^2=25\), and 18 is between them, so \(\sqrt{18}\) is a little greater than 4. In fact, \(\sqrt{18}\approx4.243\). Therefore \(4-\sqrt{18}\approx4-4.243=-0.243\). This number is negative but greater than \(-1\), so it lies between \(-1\) and 0. Option A is correct. Option B would require the result to be positive, but the root is greater than 4, making the subtraction negative. Option C and option D are positive intervals and therefore cannot contain the result. Notice that \(\sqrt{18}\) is not 18; confusing a root with the number inside it causes errors. Memory cue: first decide whether the root is larger or smaller than the number being subtracted, then estimate the interval.