Representing real numbers on the number line

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.

Moving left gives ( \frac{3}{2}-\frac{11}{6}=-\frac{1}{3} ). Subtract the distance according to direction.

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.

The governing concept is the midpoint formula for two points on a number line: M = (A + B)/2. Substitute the given values carefully: M = [(-13/2) + (-3/2)]/2 = (-16/2)/2 = (-8)/2 = -4. Therefore option B is correct. A geometric check gives the same result: A = -6.5 and B = -1.5, so -4 is 2.5 units from each endpoint. Option A may arise from adding the numerators but forgetting the final division by 2. Option C results from an incorrect arithmetic operation, while option D does not represent the average of the two endpoints. The exact formula establishes a unique midpoint.

( \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.

( \sqrt{98}\approx9.899 ), so it is closest to (10). Check distance for the nearest integer.

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 governing concept is ordering real numbers written in different forms by converting them to comparable values. First, 7/4 = 1.75. Also, √3 is approximately 1.732, while the third number is already 1.76. Thus the values compare as 1.732 < 1.75 < 1.76. Consequently, the correct increasing order is √3, 7/4, 1.76, which is option A. Option B reverses the first two values. Option C places the largest value first, and option D incorrectly places 1.76 before 7/4. The approximations are sufficiently separated, so no closer calculation is needed; alternatively, √3 < 1.75 can be confirmed by squaring positive quantities.

The governing concept is locating a negative irrational number by first bounding its positive square root between consecutive integers. Since 64 < 80 < 81, taking square roots gives 8 < √80 < 9. When each part of this inequality is multiplied by -1, the inequality signs reverse, producing -9 < -√80 < -8. Therefore P lies strictly between -9 and -8, so option A is correct. Option C gives the interval for the positive number √80 rather than P. Option B places P too close to zero, while option D places it below -9. The sign reversal is essential because negative numbers are ordered oppositely when multiplied by -1.