Representing real numbers on the number line

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

Moving right means addition, so the coordinate is (1+\sqrt{6}). Choose the sign by checking the direction.

Correct answer: A. It lies between (-4) and (-3)

To locate \(\frac{-13}{4}\), divide 13 by 4 and retain the negative sign. Since \(13\div4=3.25\), the number is \(-3.25\). On the number line, \(-3.25\) lies to the left of \(-3\) but to the right of \(-4\). Therefore it lies between \(-4\) and \(-3\), making option A correct.

The negative sign is important: negative numbers appear on the left side of zero, and a more negative number is farther left. We can also compare using fourths: \(-4=-16/4\) and \(-3=-12/4\), while \(-13/4\) is between them. It is not between \(-3\) and \(-2\), nor in the positive intervals.

Correct answer: A. (a,b,c)

Since (2<3<5), ( \sqrt{2}<\sqrt{3}<\sqrt{5}). For positive square roots, order follows the radicand.

Correct answer: A. (6.7)

Since (6^2=36) and (7^2=49), ( \sqrt{45}) is about (6.7). Estimate using nearby perfect squares.

Correct answer: A. ( -1.8)

The smallest number is farthest left, and (-1.8) is the smallest. For negative decimals, larger magnitude means farther left.

Correct answer: A. 2 and 3

Since \(3^2<12<4^2\), we have \(3<\sqrt{12}<4\). Subtracting 1 from all parts gives \(2<\sqrt{12}-1<3\), so the expression lies between 2 and 3. Option B describes the interval for \(\sqrt{12}\), not for the complete expression. Exam tip: first bound the square root between consecutive integers, then apply the remaining operation to all parts of the inequality.

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

The midpoint of \(a\) and \(b\) is \(\frac{a+b}{2}\).
So midpoint of \(0\) and \(1\) is \(\frac{0+1}{2}=\frac{1}{2}\).

Correct answer: A. \(6\)

The distance between two points on a number line is the absolute value of the difference of their coordinates: \(|3.6-(-2.4)|=|6|=6\). Therefore, the length of AB is \(6\). Remember that distance is always non-negative; \(1.2\) results from incorrectly subtracting the decimal parts only.

Correct answer: A. It is at the seventh of eight equal parts from (0) to (1)

( \frac{7}{8}) means (7) parts out of (8) equal parts from (0) to (1). The denominator gives the number of equal parts.

Correct answer: A. (10)

(99) is very close to (100), so ( \sqrt{99}) is about (10). The nearest perfect square gives a quick answer.

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

( |x-2|=3) means the distance of (x) from (2) is (3), so (x=-1) or (x=5). Check both directions in distance questions.

Correct answer: A. It lies between (3) and (4)

( \sqrt{2}\approx1.414) and ( \sqrt{3}\approx1.732), so the sum is about (3.146). Estimation is a safe method for such sums.

Correct answer: A. ( -1)

The distance from ( -\frac{5}{6}) to (-1) is ( \frac{1}{6}), and to (0) is ( \frac{5}{6}). Closeness depends on the smaller distance.

Correct answer: A. (2.9)

( \sqrt{8}\approx2.828), so (2.9) lies between it and (3). First estimate the irrational number.

Correct answer: A. (5)

( \sqrt{20}\approx4.47) and ( \sqrt{27}\approx5.19), so (5) lies between them. Use perfect squares to identify bounds.

Correct answer: A. (70)

The direct answer is option A, 70. The statement means \\(8<\\sqrt n<9\\). All quantities here are nonnegative, so squaring preserves the order: \\(8^2<n<9^2\\), hence \\(64<n<81\\). Among the choices, 70 lies strictly between 64 and 81, so it is possible. Option A works. Option B, 60, is below 64, so its square root is less than 8. Option C, 81, gives \\(\\sqrt{81}=9\\), but the number must be strictly less than 9. Option D, 100, gives \\(\\sqrt{100}=10\\), outside the interval. The endpoint symbols matter: “between 8 and 9” here excludes both endpoints. Square positive bounds only after checking that the quantities are nonnegative.

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

( \frac{3}{7}\approx0.428), ( \frac{1}{2}=0.5), and ( \frac{4}{7}\approx0.571). Use decimals or cross multiplication to compare fractions.

Correct answer: A. (p<q)

( \sqrt{11}\approx3.316), so (p\approx-3.316), which is less than (-3). Be careful with direction while comparing negatives.

Correct answer: A. \(\frac{1}{3}\)

The point to the right of \(-2\), at a distance of \(\frac{7}{3}\), is found by adding: \(-2+\frac{7}{3}=-\frac{6}{3}+\frac{7}{3}=\frac{1}{3}\). Therefore, the correct answer is \(\frac{1}{3}\).

Correct answer: A. (1.999) and (2.001)

Both (1.999) and (2.001) are at distance (0.001) from (2). Points at equal distance are equally close.

Correct answer: A. ( \sqrt{1^2+1^2})

With perpendicular sides (1) and (1), the hypotenuse is ( \sqrt{2}). Understand the construction using Pythagoras theorem.

Correct answer: A. Irrational number

(0.4040040004\ldots) is non-terminating and non-repeating, so it is irrational. Observe the decimal pattern carefully.

Correct answer: A. ( \sqrt{3}<\frac{7}{4})

( \sqrt{3}\approx1.732) and ( \frac{7}{4}=1.75), so ( \sqrt{3}) is slightly smaller. Use decimal comparison for close values.

Correct answer: A. 0.07

Since \(7^2=49\) and \(8^2=64\), \(\sqrt{50}\) is just slightly greater than 7. More precisely, \(\sqrt{50}\approx7.071\), so \(a-7\approx0.071\), which is closest to 0.07. The value 0.7 is ten times larger than this difference. Exam tip: estimate the square root near the nearest perfect square before performing the required operation.

Correct answer: A. ( -1.35)

( -\sqrt{2}\approx-1.414), so (-1.35) lies between it and (-1.3). Read order carefully in negative intervals.