Representing real numbers on the number line

Correct answer: B. Between ( -1 ) and (0)

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

Correct answer: B. 12

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.

Correct answer: A. \(-1.3\)

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.

Correct answer: A. \(\frac{12}{5}\)

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.

Correct answer: B. ( \sqrt{15}<3.88 )

The correct answer is option B: \(\sqrt{15}<3.88\). To compare carefully, square both positive numbers. We have \(3.88^2=15.0544\), while \((\sqrt{15})^2=15\). Since both numbers are positive and \(15<15.0544\), it follows that \(\sqrt{15}<3.88\). Indeed, \(\sqrt{15}\approx3.87298\). Option A reverses the true inequality. Option B is correct because the square-root value is slightly smaller. Option C is wrong because 3.88 is not the exact square root of 15; its square is 15.0544, not 15. Option D says 3.88 is less than 3, which is plainly false because 3.88 is greater than 3. On a number line, \(\sqrt{15}\) lies just to the left of 3.88. Memory cue: for positive numbers, squaring preserves order, so close square-root comparisons can be checked by squaring.

Correct answer: C. (2-\sqrt{19})

The direct answer is option C, \\(2-\\sqrt{19}\\). On a number line, moving left means subtracting the distance from the starting coordinate. Starting at 2 and moving \\(\\sqrt{19}\\) units left gives \\(2-\\sqrt{19}\\). Option C is therefore correct. Option A, \\(2+\\sqrt{19}\\), represents moving right, not left. Option B, \\(\\sqrt{19}-2\\), reverses the subtraction and does not describe the coordinate obtained from 2. Option D, \\(-2-\\sqrt{19}\\), starts from -2 rather than from 2, so it moves from the wrong point. Since \\(\\sqrt{19}\\approx4.36\\), the coordinate is about -2.36, which confirms that it is to the left of 2. Direction decides the sign: left subtracts, right adds.

Correct answer: A. ( -3.20 )

( -\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.

Correct answer: A. (2\sqrt{3})

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

Correct answer: B. Irrational number

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

Correct answer: B. \(52\)

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.

Correct answer: C. ( -4 ) and ( -3 )

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

Correct answer: B. −0.25

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.

Correct answer: A. \(\sqrt{10}\)

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.

Correct answer: B. 10 and 11

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.

Correct answer: A. Between −6 and −5

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.

Correct answer: A. ( \frac{31}{50} )

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

Correct answer: A. (11.96)

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

Correct answer: C. c

√17 ≈ 4.1231 and 33/8 = 4.125. Since 4.13 > 4.125 > 4.1231, c is the greatest and therefore lies farthest to the right on the number line.

Correct answer: B. (7.5<\sqrt{57}<7.6)

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

Correct answer: A. \(1\)

Evaluate the square root first: \(\sqrt{121}=11\). Thus, \(\frac{\sqrt{121}-4}{7}=\frac{11-4}{7}=\frac{7}{7}=1\). Therefore, the corresponding coordinate on the number line is \(1\). Option D results from dividing only 4 by 7 and ignores the subtraction involving \(\sqrt{121}\). Exam tip: follow the order of operations—evaluate the square root, simplify the numerator, and then divide.

Correct answer: A. ( -\frac{8}{5} )

Moving ( \frac{7}{5} ) to the right of (-3) gives ( -3+\frac{7}{5}=-\frac{8}{5} ). Use the given interval to choose direction.

Correct answer: A. \(\frac{13}{15}\)

Since \(169=13^2\) and \(225=15^2\), \(\sqrt{\frac{169}{225}}=\frac{\sqrt{169}}{\sqrt{225}}=\frac{13}{15}\). The principal square root is non-negative, so the positive values 13 and 15 are used. Option B is the original fraction, not its square root. Exam tip: For a fraction whose numerator and denominator are perfect squares, take the positive square root of each separately.

Correct answer: A. (2.90)

( \sqrt{8}+\frac{1}{10}\approx2.928 ). Therefore (2.90) does not satisfy it and no given value satisfies the condition.

Correct answer: A. Point A

C = −43/4 = −10.75, so B and C are at the same position. Since √116 ≈ 10.7703, A ≈ −10.7703. This is less than −10.75, so A lies farthest to the left on the number line.

Correct answer: B. (4-\sqrt{6}>\frac{31}{20})

(4-\sqrt{6}\approx1.551) and ( \frac{31}{20}=1.55 ), so the first value is slightly greater. Estimate close values accurately.