Representing real numbers on the number line

( \sqrt{31}\approx5.568 ), so (5-\sqrt{31}\approx-0.568). Always check the sign in root subtraction.

Since \(12^2=144<145<169=13^2\), we get \(12<\sqrt{145}<13\). Comparing this with \(m<\sqrt{145}<m+1\) gives \(m=12\). Option 13 is incorrect because \(\sqrt{145}<13\), so \(13<\sqrt{145}\) is not true. Exam tip: the integer part of a square root is the greatest integer whose square is less than or equal to the given number.

Since \(1.3^2=1.69\), we have \(\sqrt{1.69}=1.3\). The negative sign outside the radical makes the result negative, so \(-\sqrt{1.69}=-1.3\). Option B ignores the negative sign, while option D gives the radicand itself instead of its square root. Exam tip: find the square root first and then apply any sign outside the radical.

Since \(\sqrt{144}=12\), we get \(x=\frac{12}{5}\). Therefore, \(x\) is represented by the point \(\frac{12}{5}\) on the number line. Exam tip: simplify the square root first and then divide by 5; \(\frac{144}{5}\) results from skipping the square-root step.

( \sqrt{15}\approx3.873 ), which is slightly less than (3.88). Use more accurate estimation for close values.

Moving left means subtracting the distance, so the coordinate is (2-\sqrt{19}). Choose the sign by direction.

The governing concept is that the midpoint of two real-number points is their arithmetic average. Let the endpoints be 5/12 and 7/12. Then M = [(5/12) + (7/12)]/2 = (12/12)/2 = 1/2. Therefore option B is correct. Because the fractions have the same denominator, their sum is especially direct: the numerators add to 12, giving 1, and dividing by 2 gives 1/2. A geometric check also works: 5/12 = 0.4167 and 7/12 = 0.5833, both exactly 1/12 away from 1/2. Option A is left of the first endpoint, while options C and D are right of the second or too far away.

( -\sqrt{10}\approx-3.162 ) and ( -\frac{19}{6}\approx-3.167 ). (-3.20) is not between them, so recheck direction for close negative values.

( \sqrt{75}=5\sqrt{3} ) and ( \sqrt{27}=3\sqrt{3} ), so the difference is (2\sqrt{3}). Subtract only like radicals.

This decimal is non-terminating and non-repeating, so it is irrational. Check carefully whether the pattern repeats or not.

Since \(7.2<\sqrt{n}<7.3\) and both boundary values are positive, squaring gives \(7.2^2<n<7.3^2\). Thus, \(51.84<n<53.29\). Only \(52\) lies in this interval; \(51\) and \(50\) are too small, while \(54\) is too large. Exam tip: when both sides of an inequality are positive, squaring preserves its direction.

( -\frac{31}{9}\approx-3.444 ), so it lies between (-4) and (-3). Convert negative fractions to decimals.

The coordinate of the midpoint is the average of the coordinates of the two endpoints: \(\frac{-3.25+2.75}{2}=\frac{-0.50}{2}=-0.25\). Therefore, option B is correct. Option A is only the sum of the two coordinates, not their average. In the exam, remember to divide the sum by 2 when finding a midpoint.

Since \(9<10<16\), we get \(3<\sqrt{10}<4\). As 10 is not a perfect square, \(\sqrt{10}\) is irrational. \(\frac{13}{4}\) lies in the interval but is rational. Exam tip: compare nearby perfect squares.

Since \(9^2=81<89<100=10^2\), we get \(9<\sqrt{89}<10\). Adding 1 to all parts gives \(10<\sqrt{89}+1<11\), so the expression lies between 10 and 11. Option A bounds only \(\sqrt{89}\), not the complete expression. Exam tip: compare the number with the nearest consecutive perfect squares to bound a square root.

Since \(8^2<68<9^2\), we have \(8<\sqrt{68}<9\). More precisely, \(\sqrt{68}\approx8.246\), so \(p=3-8.246\approx-5.246\). Therefore, \(p\) lies between −6 and −5. Option B is incorrect because \(p\) is less than −5. Exam tip: For an expression involving a negative square root, estimate the square root first and then perform the subtraction carefully.

Moving left gives ( \frac{9}{10}-\frac{7}{25}=\frac{31}{50} ). Subtract the distance according to direction.

( \sqrt{143}\approx11.96 ) because (11.96^2) is close to (143). Check nearby squares for large roots.

On converting the values to decimals, \(\sqrt{17}\approx 4.1231\), \(\frac{33}{8}=4.125\), and \(c=4.13\). Thus, \(4.13>4.125>4.1231\), so \(c\) is farthest to the right on the number line. The closest distractor is \(b\), but \(4.13\) is greater than \(4.125\). Exam tip: Convert irrational and fractional values to comparable decimal forms before ordering them.

(7.5^2=56.25) and (7.6^2=57.76), so ( \sqrt{57} ) lies between them. Check squares for decimal bounds.