Representing real numbers on the number line

\(\frac{17}{5}=3.4\) and \(\sqrt{12}\approx3.464\). Thus, \(3.4<3.45<3.464\), so 3.45 is the correct value. Options 3.47 and 3.50 are greater than \(\sqrt{12}\), while 3.30 is less than \(\frac{17}{5}\). Exam tip: Convert the fraction to a decimal and estimate the square root before comparing the values.

( -\sqrt{18}-2\approx-6.243 ), so it lies between (-7) and (-6). Estimate negative sums carefully.

( \sqrt{192}=8\sqrt{3} ) and ( \sqrt{75}=5\sqrt{3} ), so the difference is (3\sqrt{3}). Simplify the radicals first.

( \sqrt{126}\approx11.22 ) and ( \sqrt{80}\approx8.94 ), so the difference is about (2.28). Estimate both square roots first.

( |x-1.5|=4.25 ) means the distance of (x) from (1.5) is (4.25). Moving both directions gives ( -2.75 ) and (5.75).

( \frac{41}{10}=4.1 ), ( \sqrt{17}\approx4.123 ), and (4.13). Convert close values to decimals for comparison.

( -\sqrt{93}\approx-9.644 ) and (-9.62) is greater. The greater number lies farther right on the number line.

Since \(2.8^2=2.8\times2.8=7.84\), we have \(\sqrt{7.84}=2.8\). The nearby choices are incorrect because \(2.7^2=7.29\) and \(2.9^2=8.41\). In an exam, squaring the options is a quick way to verify a decimal square root.

The distance is \( \left|\frac{9}{10}-\left(-\frac{17}{5}\right)\right|=\frac{43}{10} \). Use absolute value while finding distance.

( \sqrt{48}\approx6.928 ) and ( \sqrt{49}=7 ), so (6.95) lies between them. Use accurate estimation for close roots.

( \sqrt{39}\approx6.245 ), so (6-\sqrt{39}\approx-0.245). Always check the sign in root subtraction.

Since \(2.56=(1.6)^2\), we have \(\sqrt{2.56}=1.6\). The negative sign outside the radical changes the entire value to negative, so \(-\sqrt{2.56}=-1.6\). Option B omits the negative sign, while C and D result from misreading the decimal or the radicand as the square root. Exam tip: find the principal square root first, then apply any sign outside the radical.

Since \(169=13^2\), we have \(\sqrt{169}=13\). Therefore, \(x=\frac{13}{6}\), so this is the point represented on the number line. Option B incorrectly leaves 169 in the numerator, while option C reverses the fraction. Exam tip: simplify the square root first and then perform the remaining operation.

( \sqrt{21}\approx4.583 ), which is slightly less than (4.59). Use more accurate estimation for close values.

Moving left means subtracting the distance, so the coordinate is ( -4-\sqrt{23} ). Choose the sign by direction.

The midpoint is ( \frac{\frac{7}{15}+\frac{11}{15}}{2}=\frac{3}{5} ). Use the average of the two values for the midpoint.

The governing concept is representation and comparison of real numbers on a number line, especially the order of negative numbers. Estimate the endpoints: √12 is approximately 3.4641, so −√12 is approximately −3.4641. Also, −17/5 = −3.4 exactly. Therefore a number between them must be greater than −3.4641 but less than −3.4. Option B, −3.43, satisfies −3.4641 < −3.43 < −3.4. Option A, −3.30, and option D, −3.20, are to the right of −3.4, while option C, −3.60, is to the left of −3.4641. The comparison may seem counterintuitive because a negative number closer to zero is larger.

( \sqrt{108}=6\sqrt{3} ) and ( \sqrt{48}=4\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 whether the decimal pattern truly repeats or not.

Since 8.4 and 8.5 are positive, squaring the inequality gives c(8.4c)^2<n<c(8.5c)^2. Thus, 70.56<n<72.25, and among the given options only 71 lies in this interval. 70 is below the interval and 73 is above it. Exam tip: when a square root lies between positive numbers, square both endpoints to find the range of the number under the root.