Representing real numbers on the number line

Correct answer: B. (1) and (2)

( \sqrt{31}\approx5.57 ) and ( \sqrt{12}\approx3.46 ) so the difference is about (2.11). Estimate both roots first.

Correct answer: A. ( -0.5 ) and ( -5.5 )

( |x+3|=2.5 ) means the distance of (x) from (-3) is (2.5). Moving both ways gives (-0.5) and (-5.5).

Correct answer: A. (Q)

( -\sqrt{20}\approx-4.472 ) and (-4.4) is greater than it. The greater number lies to the right on a number line.

Correct answer: A. 1.7

The principal square root is nonnegative, and 1.7 × 1.7 = 2.89. Therefore, √2.89 = 1.7, so it is marked at coordinate 1.7.

Correct answer: A. \( \frac{19}{6} \)

The distance is \( \left|\frac{5}{6}-\left(-\frac{7}{3}\right)\right|=\frac{19}{6} \). Always use absolute value for distance.

Correct answer: A. 4.3

\(\sqrt{18}\approx 4.243\) and \(\sqrt{19}\approx 4.359\). Since \(4.243<4.3<4.359\), the value 4.3 lies between the two square roots. The value 4.1 is less than \(\sqrt{18}\), while 4.5 is greater than \(\sqrt{19}\). In the exam, estimate the decimal values of the relevant square roots to compare them quickly.

Correct answer: A. Between (-2) and (-1)

( \sqrt{10}\approx3.162 ), so (2-\sqrt{10}\approx-1.162). Estimation is important in root subtraction.

Correct answer: A. 7

Since \(7^2=49<57<64=8^2\), we have \(7<\sqrt{57}<8\). Comparing this with \(m<\sqrt{57}<m+1\) gives \(m=7\). Option 6 is too small because \(6^2=36<57\), while 8 is an upper bound, not the required integer. Exam tip: locate a square root between two consecutive perfect squares to find its integer part.

Correct answer: A. \(-0.6\)

\(\sqrt{0.36}=0.6\) because \(0.6^2=0.36\). The negative sign outside the radical makes the entire value negative, so \(-\sqrt{0.36}=-0.6\). Option B ignores the outside negative sign, while C and D have incorrect decimal values. Exam tip: first evaluate the principal square root, then apply any sign outside the radical.

Correct answer: A. \(\frac{9}{4}\)

Since \(\sqrt{81}=9\), we get \(x=\frac{9}{4}\). Option B incorrectly uses 81 instead of its square root, while options C and D do not result from the given expression. In the exam, simplify the square root first and then perform the division.

Correct answer: A. ( \sqrt{5}<2.24 )

( \sqrt{5}\approx2.236 ), which is slightly less than (2.24). Compare close decimals to more places.

Correct answer: A. \(-1-\sqrt{6}\)

Moving left on the number line means subtracting the given distance from the starting coordinate. Hence, \(C=-1-\sqrt{6}\). Option B adds the distance and therefore represents a point to the right of \(-1\). Exam tip: subtract distances when moving left and add them when moving right.

Correct answer: A. ( \frac{1}{2} )

The point exactly midway between two numbers is their arithmetic mean. For \(\frac25\) and \(\frac35\), add them and divide by 2: \(\frac{\frac25+\frac35}{2}=\frac{\frac55}{2}=\frac12\). Therefore the midpoint is \(\frac12\), so option A is correct.

This also makes sense geometrically. The two endpoints are 0.4 and 0.6 on the number line, and 0.5 is equally distant from each: the distance from 0.4 to 0.5 is 0.1, as is the distance from 0.5 to 0.6. The values \(\frac52\), \(\frac15\), and \(\frac45\) do not have equal distances from both endpoints.

Correct answer: A. ( \sqrt{3} )

( \sqrt{27}=3\sqrt{3} ) and ( \sqrt{12}=2\sqrt{3} ), so the difference is ( \sqrt{3} ). Subtract like radicals.

Correct answer: A. Irrational number

The decimal expansion is non-terminating and non-repeating because the number of 7s after each 2 keeps increasing. Therefore, it cannot be expressed in the form p/q and is irrational. A rational number has a terminating or recurring decimal expansion, so option B is incorrect. Exam tip: a non-terminating, non-repeating decimal is irrational.

Correct answer: A. 42

Since \(6.4<\sqrt{n}<6.5\), squaring both sides gives \(6.4^2<n<6.5^2\). Therefore, \(40.96<n<42.25\). Among the given choices, only 42 lies in this interval. The values 40, 45 and 49 fall outside it. Exam tip: for positive quantities, squaring preserves the direction of the inequality.

Correct answer: A. −4 and −3

\(-\frac{17}{5}=-3.4\). On the number line, \(-3.4\) lies to the right of \(-4\) and to the left of \(-3\), so it lies between \(-4\) and \(-3\). Exam tip: for negative numbers, the number with the greater magnitude is farther to the left on the number line.

Correct answer: A. -0.75

The midpoint of two points \(x_1\) and \(x_2\) is \(\frac{x_1+x_2}{2}\). Therefore, \(\frac{-2.75+1.25}{2}=\frac{-1.50}{2}=-0.75\), so option A is correct. Option B is only the sum of the two coordinates, not their midpoint. Exam tip: always divide the sum of the coordinates by 2 to find a midpoint.

Correct answer: A. 0.09

The principal square root is non-negative, and 0.09 × 0.09 = 0.0081. Therefore, √0.0081 = 0.09, so the point is at 0.09 on the number line.

Correct answer: A. 7 and 8

Since \(6^2=36<37<49=7^2\), we get \(6<\sqrt{37}<7\). Adding 1 to all parts gives \(7<\sqrt{37}+1<8\), so the expression lies between 7 and 8. Exam tip: To bound a square root, compare the number with the consecutive perfect squares around it. Option B is incorrect because it represents the interval for \(\sqrt{37}\), not for \(\sqrt{37}+1\).

Correct answer: A. Between \(-5\) and \(-4\)

Since \(6<\sqrt{45}<7\), multiplying by \(-1\) reverses the inequalities: \(-7< -\sqrt{45}< -6\). Adding 2 throughout gives \(-5<p<-4\). Therefore, \(p\) lies between \(-5\) and \(-4\). Option B is incorrect because \(p\) is less than \(-4\). Exam tip: Compare a square root with neighbouring perfect squares instead of calculating its decimal value.

Correct answer: A. a

Here, b = 14/5 = 2.8 and c = 2.81, while √8 ≈ 2.828. Thus, 2.828 > 2.81 > 2.8, so a is greatest.

Correct answer: A. \(4.8<\sqrt{24}<4.9\)

Since \(4.8^2=23.04\) and \(4.9^2=24.01\), and \(23.04<24<24.01\), it follows that \(4.8<\sqrt{24}<4.9\). Options B and D give intervals that are too small, while both endpoints in option C have squares greater than 24. Exam tip: To verify a decimal bound for a square root, compare the squares of its endpoints with the given number.

Correct answer: A. 1

First evaluate the square root: \(\sqrt{64}=8\). Therefore, \(\frac{\sqrt{64}-3}{5}=\frac{8-3}{5}=\frac{5}{5}=1\). Hence, the expression represents the point 1 on the number line. Option D results from dividing 3 by 5 without correctly evaluating the numerator; evaluate the square root and subtraction before dividing.

Correct answer: A. ( -\frac{3}{4} )

Moving ( \frac{5}{4} ) to the right of (-2) gives ( -2+\frac{5}{4}=-\frac{3}{4} ). Use the given interval to choose the direction.