Representing real numbers on the number line

Correct answer: A. Because (2^2<7<3^2)

Since (2^2=4) and (3^2=9), (\sqrt{7}) lies between (2) and (3). In exams, compare squares first.

Correct answer: A. (1.4<\sqrt{2}<1.5)

Because (1.4^2=1.96) and (1.5^2=2.25), (\sqrt{2}) lies between them. For hard questions, use squares of decimals.

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

Since (2^2<5<3^2), (\sqrt{5}) lies between (2) and (3), so (-\sqrt{5}) lies between (-3) and (-2). On the negative side, order reverses.

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

The midpoint of (0) and (1) is (\frac{0+1}{2}=\frac{1}{2}). To find a midpoint on a number line, take the average.

Correct answer: A. (8) parts

Since (2=\frac{8}{4}), the interval from (0) to (2) has (8) fourth-parts and (\frac{7}{4}) is at the seventh part. Use the denominator to make equal units.

Correct answer: A. 3 and 4

Since \(3^2=9\) and \(4^2=16\), we have \(9<10<16\). Taking square roots gives \(3<\sqrt{10}<4\), so \(\sqrt{10}\) lies between 3 and 4. Option B is incorrect because \(\sqrt{10}>\sqrt{9}=3\). Exam tip: To locate a square root, identify the consecutive perfect squares between which its radicand lies.

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

The direct answer is option A, between \(-3\) and \(-2\). First convert the fraction: \(-\frac{11}{5}=-2.2\). On the number line, \(-3<-2.2<-2\), so the number lies between the consecutive integers \(-3\) and \(-2\). Option A is correct. Option B, \(-2 ext{ and }-1\), is wrong because \(-2.2 ext{ is less than }-2\), not greater. Option C, 2 and 3, describes positive numbers, while the given fraction is negative. Option D, \(-1 ext{ and }0\), is also too far to the right. A common mistake is thinking that 2.2 lies between 2 and 3 and forgetting that the minus sign reverses the location. On a negative number line, the more negative number is farther left.

Correct answer: A. (-3.5) and (3.5)

Distance from (0) is (|x|), so (|x|=3.5) gives (x=\pm3.5). Distance is always a positive measure.

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

Moving right means addition, so (-1+\frac{7}{3}=\frac{4}{3}). Decide addition or subtraction by direction.

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

Because ((\sqrt{2})^2+1^2=3), the hypotenuse is (\sqrt{3}). Successive square roots are constructed using Pythagoras.

Correct answer: A. (1.625)

(\frac{13}{8}=1.625), so both represent the same point. Convert the fraction into a decimal to identify position.

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

Since \(12=4\\times3\) and \(\\sqrt{4}=2\), we get \(\\sqrt{12}=\\sqrt{4\\times3}=2\\sqrt{3}\). Therefore, option A is correct. The squares of \(3\\sqrt{2}\), \(4\\sqrt{3}\), and \(\\sqrt{6}\) are 18, 48, and 6 respectively, so none of them equals \(\\sqrt{12}\). Exam tip: when simplifying a surd, first extract the greatest perfect-square factor from under the radical sign.

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

(\frac{\sqrt{2}}{2}) is irrational and its value lies between (0) and (1). An irrational divided by a non-zero rational remains irrational.

Correct answer: A. (-\sqrt{3},-\frac{3}{2},0,\sqrt{2})

Since (-\sqrt{3}\approx-1.732) and (-\frac{3}{2}=-1.5), (-\sqrt{3}) is farther left. For negative numbers, larger magnitude means farther left.

Correct answer: A. (A)

Since (-2.75<-2.7), (A) is to the left on the number line. In negative decimals, the more negative number is farther left.

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

Points at the same distance from (0) occur on both sides, so the values are (\pm\sqrt{13}). In distance questions, check both directions.

Correct answer: A. ((4,5))

Because (4^2=16) and (5^2=25), (4<\sqrt{20}<5). Perfect-square bounds quickly give the interval.

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

Let \(x=0.333\ldots\). Then \(10x=3.333\ldots\). Subtracting the first equation from the second gives \(9x=3\), so \(x=\frac{1}{3}\). Options B, C and D represent the terminating decimals \(0.3\), \(0.33\) and \(0.333\), respectively, not the infinitely repeating decimal. Exam tip: To convert a repeating decimal into a fraction, assign it to \(x\), multiply by the appropriate power of \(10\), and subtract.

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

(\sqrt{3}\approx1.732), so it lies near (1.732) on the number line. Remember approximations of common square roots.

Correct answer: A. (1)

The midpoint is (\frac{\frac{5}{6}+\frac{7}{6}}{2}=1). For fractions, the average method is safest.

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

Since \\(1<\\sqrt{3}<2\\), subtracting \\(\\sqrt{3}\\) from 2 gives \\(0<2-\\sqrt{3}<1\\). Therefore, the number lies in \\((0,1)\\). Its approximate value is 0.268, so the interval \\((1,2)\\) is not possible. Exam tip: estimate the irrational term first, then locate the resulting value.

Correct answer: A. \(\frac{5}{4}\)

Since \(1.25\) has two digits after the decimal point, \(1.25=\frac{125}{100}\). Dividing the numerator and denominator by 25 gives \(\frac{125}{100}=\frac{5}{4}\). Option B, \(\frac{4}{5}\), is the reciprocal of \(\frac{5}{4}\), not its equivalent fraction. Exam tip: convert a terminating decimal to a fraction with denominator \(10^n\), then reduce it to the lowest terms.

Correct answer: A. (2.45)

The direct answer is option A, 2.45. We know \(\sqrt6\approx2.449489\). Compare each candidate with this value. For A, the difference is about \(|2.45-2.449489|\approx0.000511\), which is extremely small. For B, 2.10, the difference is about 0.3495. For C, 2.90, the difference is about 0.4505. For D, 3.20, the difference is about 0.7505. Therefore A is clearly closest. Option B is too low, while C and D are too high and farther away. A useful check is that \(2.45^2=6.0025\), very close to 6, so 2.45 is a strong approximation of \(\sqrt6\). Memory cue: the closest number has the smallest absolute difference, not merely the same rough size.

Correct answer: A. (1+\frac{2}{7})

(\frac{9}{7}=1+\frac{2}{7}), so choose the second seventh part after (1). Converting an improper fraction into a mixed form is useful.

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

The midpoint is (\frac{-\frac{5}{2}-\frac{1}{2}}{2}=-\frac{3}{2}). Pay attention to signs in negative fractions.