Representing real numbers on the number line

Correct answer: A. (30)

From (5<\sqrt{n}<6), we get (25<n<36), so (30) is possible. Square positive sides in square-root inequalities.

Correct answer: A. ((1,2))

Since (2<\sqrt{5}<3), (1<-1+\sqrt{5}<2). When adding or subtracting a constant, adjust the whole inequality.

Correct answer: A. (4.24)

The direct answer is option A, 4.24. Simplify the radical: \(\sqrt{18}=\sqrt{9\cdot2}=3\sqrt{2}\). Since \(\sqrt{2}\approx1.4142\), we get \(3\sqrt{2}\approx4.2426\), which is nearest to 4.24. We can also check that \(4.24^2\) is close to 18. Option A, 4.24, is correct because its difference from 4.2426 is very small. Option B, 3.24, is about one unit too small. Option C, 5.24, is about one unit too large. Option D, 6.24, is much farther away. The nearby perfect squares 16 and 25 show that \(\sqrt{18}\) lies between 4 and 5, and the calculation gives its precise approximate position. Do not confuse \(\sqrt{18}\) with 18 or with \(\sqrt{9}+\sqrt{2}\); radicals cannot generally be split across addition.

Correct answer: A. (\frac{3}{5},\frac{2}{3},\frac{5}{6})

With common denominator (30), (\frac{18}{30}<\frac{20}{30}<\frac{25}{30}). Use a common denominator to order fractions.

Correct answer: A. ((3,4))

(\sqrt{2}\approx1.414) and (\sqrt{3}\approx1.732), so the sum is about (3.146). Add approximate values for sums.

Correct answer: A. (4)

The direct answer is option A, 4. Length on a number line is the absolute difference between the coordinates. Substitute the given values: \(PQ=\left|\frac{11}{2}-\frac{3}{2}\right|=\left|\frac{8}{2}\right|=|4|=4\). Option A is correct. Option B, \(\frac72\), results from an incorrect numerator subtraction and is 3.5, not the actual gap. Option C, 5, is the coordinate of Q, not the distance from P to Q. Option D, \(\frac{11}{2}\), is again just Q's coordinate, not the length of the segment. Since Q is to the right of P, we may also subtract directly without changing the sign, but absolute value is always safe. Exam cue: distance is not an endpoint coordinate; it is the difference between endpoints.

Correct answer: A. (\sqrt{13})

Because (11<13<15), (\sqrt{11}<\sqrt{13}<\sqrt{15}). For positive square roots, the order of radicands is preserved.

Correct answer: A. (-1) and (5)

(|x-2|=3) means (x) is (3) units away from (2), so (x=-1,5). Place the distance on both sides of the center.

Correct answer: A. ((-4,-3))

The direct answer is option A: \(-\sqrt{12}\) lies in \((-4,-3)\). Since \(9<12<16\), taking square roots gives \(3<\sqrt{12}<4\). Multiplying the whole inequality by \(-1\) reverses the signs: \(-3>-\sqrt{12}>-4\), which can be written as \(-4< -\sqrt{12}< -3\). Option A is correct. Option B, \((-3,-2)\), is too close to zero; the number is approximately \(-3.46\). Option C is positive, but \(-\sqrt{12}\) is negative. Option D, \((-5,-4)\), is too far left because the value is greater than \(-4\). Common mistake: when multiplying an inequality by a negative number, reverse its direction.

Correct answer: A. It is irrational between (0) and (1)

This decimal is non-terminating and non-repeating, so it is irrational and lies between (0) and (1). Identify rationality by the decimal pattern.

Correct answer: A. 3.1

The principal square root is positive. Since \(3.1^2=3.1\times3.1=9.61\), it follows that \(\sqrt{9.61}=3.1\). Option B is incorrect because \(3.01^2=9.0601\), not \(9.61\). In the exam, verify a decimal square by multiplying the number by itself.

Correct answer: A. x

Both numbers are positive, so their squares can be compared. \(x^2=\left(\frac{17}{10}\right)^2=\frac{289}{100}=2.89\), whereas \(y^2=(\sqrt{3})^2=3\). Thus, \(x^2<y^2\) implies \(x<y\), so \(x\) lies to the left on the number line. The distractor \(y\) is incorrect because \(y\approx1.732\), while \(x=1.7\). Exam tip: For positive numbers, comparing their squares is often easier than comparing the numbers directly.

Correct answer: A. Left ( \frac{7}{4} ) units

A negative number lies to the left of (0), and its distance is its absolute value (\frac{7}{4}). Identify direction and distance separately.

Correct answer: A. (3)

The direct answer is option A: \(3\). Evaluate each perfect square first: \(\sqrt{25}=5\) and \(\sqrt{4}=2\). Therefore \(\sqrt{25}-\sqrt{4}=5-2=3\). Option A is correct. Option B, 7, would result from adding 5 and 2, not subtracting them. Option C, 21, does not follow from either the square roots or their difference. Option D, \(\sqrt{21}\), incorrectly combines the radicands; in general, \(\sqrt{25}-\sqrt{4}\ne\sqrt{21}\). The number 3 is represented three units to the right of zero on the number line. Memory cue: simplify perfect-square roots before performing the operation.

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

(\sqrt{8}=2\sqrt{2}), so the midpoint is (\frac{\sqrt{2}+2\sqrt{2}}{2}=\frac{3\sqrt{2}}{2}). Simplify first, then average.

Correct answer: A. -(\frac{5}{6}), (-\frac{3}{4}), (-\frac{2}{3}) / (-\frac{5}{6}), (-\frac{3}{4}), (-\frac{2}{3})

For negative numbers, the one with larger magnitude is farther left, so (-\frac{5}{6}<-\frac{3}{4}<-\frac{2}{3}). Compare positive values first, then reverse the order.

Correct answer: A. 7.07

√50 = √(25 × 2) = 5√2. Using √2 ≈ 1.4142 gives √50 ≈ 5 × 1.4142 = 7.071, which is closest to 7.07. Also, 7² < 50 < 8², so the value lies between 7 and 8.

Correct answer: A. (1)

(\sqrt{9}=3), and moving left means subtracting, so (P=4-3=1). In direction questions, simplify the distance first.

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

(\pi\approx3.14) and (\sqrt{10}\approx3.16), so (\pi<\sqrt{10}). Good approximations help compare close irrationals.

Correct answer: A. (4) and (1)

Because (4^2+1^2=17), the hypotenuse will be (\sqrt{17}). Pythagoras theorem is useful for square-root construction on the number line.

Correct answer: A. ((0,1))

(\sqrt{5}\approx2.236) and (\sqrt{2}\approx1.414), so the difference is about (0.822). Use short approximations to locate differences of irrationals.

Correct answer: A. 4 and 5

Since \(4^2=16<18<25=5^2\), it follows that \(4<\sqrt{18}<5\). Therefore, \(\sqrt{18}\) lies between 4 and 5 on the number line. Option B is incorrect because \(\sqrt{18}>4\). Exam tip: compare the number with the nearest consecutive perfect squares to locate its square root.

Correct answer: B. (2.36)

The distance from (2.357) to (2.36) is (0.003), and to (2.35) is (0.007). The smaller distance gives the closer point.

Correct answer: A. ( -3) and (-2)

The direct answer is option A: between -3 and -2. Since \\(2^2=4<7<9=3^2\\), we have \\(2<\\sqrt7<3\\). Multiplying this inequality by -1 reverses both inequality signs, giving \\(-3<-\\sqrt7<-2\\). Therefore \\(-\\sqrt7\\) lies between the consecutive integers -3 and -2. Option A is correct. Option B, -2 and -1, is too close to zero; the number is approximately -2.65. Option C contains positive numbers, but \\(-\\sqrt7\\) is negative. Option D, -4 and -3, is too far left because \\(-2.65\\) is greater than -3. The key exam rule is that multiplying an inequality by a negative number reverses its signs.

Correct answer: B. (3.4)

( \frac{17}{5}=3.4 ), so the point lies between (3) and (4). Convert the fraction to a decimal to locate it.