Representing real numbers on the number line

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.

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.

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.

( |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} ).

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.

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.

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

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.

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

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.

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.

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

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

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

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

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.

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

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

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

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.

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.

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

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.

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.

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.