Representing real numbers on the number line

\(\frac{9}{4}=2.25\) and \(\sqrt{6}\approx 2.449\). Thus, 2.4 lies between 2.25 and 2.449. Option B, 2.2, is less than \(\frac{9}{4}\), while 2.5 and 3.0 are greater than \(\sqrt{6}\). Exam tip: Convert the fraction and square root to suitable decimal approximations before comparing them.

( -\sqrt{2}-1\approx-2.414 ), so it lies between (-3) and (-2). Estimate negative sums carefully.

( \sqrt{48}=4\sqrt{3} ) and ( \sqrt{27}=3\sqrt{3} ), so the difference is ( \sqrt{3} ). Simplify the radicals first.

Compare the nearest perfect squares: \(7^2=49\) and \(8^2=64\). Since \(49<52<64\), taking square roots gives \(7<\sqrt{52}<8\). Therefore, \(\sqrt{52}\) lies between 7 and 8. Exam tip: To bound a square root, identify the consecutive perfect squares immediately below and above the number.

The governing concept is bounding a square root between consecutive integers and then applying the effect of a negative sign. Since 25 < 29 < 36, taking principal square roots gives 5 < √29 < 6. Multiplying all parts by -1 reverses the inequality signs, so -6 < -√29 < -5. Therefore x lies between -6 and -5, making option B correct. Options C and D describe positive intervals and ignore the negative sign in x. Option A is also incorrect because -√29 is less than -5 but greater than -6; it is not between -5 and -4. The bounds are exact enough to identify the interval without calculating a decimal approximation.

Convert the fraction to a decimal by dividing 23 by 8: \(23 \div 8 = 2.875\). Therefore, \(\frac{23}{8}\) corresponds to the point 2.875 on the number line. Options A and D represent \(\frac{21}{8}\) and \(\frac{19}{8}\), respectively, so they are incorrect. Exam tip: Convert a fraction to a decimal or mixed number before locating it on the number line.

Moving right means adding the distance, so the coordinate is ( -3+\sqrt{14} ). Choose addition or subtraction by direction.

Since \(1.8^2=1.8\times1.8=3.24\), we have \(\sqrt{3.24}=1.8\). The square root symbol denotes the positive square root, so the corresponding point has coordinate \(1.8\). The closest distractor, \(1.6\), is incorrect because \(1.6^2=2.56\). Exam tip: verify a decimal square root by squaring the proposed value and checking its place value.

( -\frac{19}{6}\approx-3.167 ), so it lies between (-4) and (-3). Converting a negative fraction to decimal is useful.

Since \(\sqrt{6}\approx2.449\), \(c=2.48\), and \(\frac{5}{2}=2.5\), we get \(2.449<2.48<2.5\). Therefore, the increasing order is \(a<c<b\). Remember that increasing order means arranging numbers from least to greatest.

Since \(8^2=64\) and \(9^2=81\), \(\sqrt{65}\) is very close to 8. More precisely, \(\sqrt{65}\approx8.062\), which rounds to 8.1 to one decimal place. Option 7.9 lies below 8 and is not the correct nearest decimal estimate. Exam tip: bracket a square root between two consecutive perfect squares, then round its decimal value to the required place.

( \sqrt{21}\approx4.583 ) and ( \sqrt{22}\approx4.690 ), so (4.65) lies between them. Estimate accurately for close roots.

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.

( \frac{11}{12} ) means (11) parts out of (12) equal parts. The denominator gives the number of equal parts.

( |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} ).

( \sqrt{8}=2\sqrt{2} ), so ( \sqrt{2}+\sqrt{8}=3\sqrt{2} ). Only like radicals can be added.

The governing concept is converting a rational number from fractional form to decimal form while preserving its sign. Dividing 7 by 8 gives 7 ÷ 8 = 0.875; equivalently, 7/8 can be written as 875/1000. Since the original fraction is negative, -7/8 = -0.875. Therefore option A is correct. Option B changes the order of the digits and has a different value. Option C has the correct magnitude but loses the negative sign, so it represents positive 7/8 and lies to the right of zero. Option D is also negative but its magnitude is 1.125, which is not equal to 0.875. Exact fraction-to-decimal conversion confirms the answer.

The midpoint is ( \frac{-\frac{5}{3}+\frac{7}{3}}{2}=\frac{1}{3} ). Take the average to find the midpoint.

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