Representing real numbers on the number line

Correct answer: A. ( -\frac{5}{2})

The midpoint is ( \frac{-\frac{9}{2}-\frac{1}{2}}{2}=-\frac{5}{2}). Add the fractions first, then divide by (2).

Correct answer: A. ( \frac{22}{7}>\pi)

( \frac{22}{7}\approx3.142857) and ( \pi\approx3.14159), so ( \frac{22}{7}) is slightly larger. Do not treat common approximations as exactly equal.

Correct answer: A. Between (3.8) and (3.9)

The direct answer is option A: between 3.8 and 3.9. To locate a positive square root, compare squares because squaring preserves order for non-negative numbers. We have \(3.8^2=14.44\) and \(3.9^2=15.21\). Since \(14.44<15<15.21\), taking square roots gives \(3.8<\sqrt{15}<3.9\). Therefore option A is correct. Option B, between 3.5 and 3.6, is wrong because their squares are 12.25 and 12.96, both below 15. Option C, between 4.1 and 4.2, is wrong because their squares, 16.81 and 17.64, are above 15. Option D, between 2.9 and 3.0, is wrong because their squares are 8.41 and 9, far below 15. The nearby whole squares also help: \(3^2=9<15<16=4^2\), so the answer must lie between 3 and 4. Exam cue: square both proposed decimal boundaries.

Correct answer: A. Between \(-1\) and \(0\)

Moving \(\sqrt{13}\) units to the right from \(-4\) gives \(x=-4+\sqrt{13}\). Since \(3^2<13<4^2\), we have \(3<\sqrt{13}<4\). Adding \(-4\) throughout gives \(-1<x<0\), so \(x\) lies between \(-1\) and \(0\). Option B is incorrect because the value of \(x\) is still negative. Exam tip: locate a square root between two consecutive integers before adding or subtracting it.

Correct answer: A. \(\frac{1}{8}\)

Write the decimal as a fraction first: \(0.125=\frac{125}{1000}\). Dividing the numerator and denominator by \(125\) gives \(\frac{125}{1000}=\frac{1}{8}\), so option A is correct. The distractor \(\frac{1}{4}\) equals \(0.25\), not \(0.125\). Exam tip: for a decimal with three digits after the decimal point, initially use \(1000\) as the denominator and then simplify.

Correct answer: A. ( \frac{1}{12})

The direct answer is option A: \(1/12\). On a number line, moving to the right means adding the distance. Starting at \(-3/4\) and moving right by \(5/6\) gives \(-3/4+5/6\). The least common denominator is 12: \(-3/4=-9/12\) and \(5/6=10/12\). Therefore \(-9/12+10/12=1/12\). Option A is correct. Option B, \(-19/12\), results from adding the magnitudes while keeping the starting negative sign, which represents movement in the wrong direction. Option C, \(5/24\), comes from an incorrect denominator or arithmetic step. Option D, \(-1/12\), would represent moving left by the corresponding difference, not moving right. The phrase 'to the right' is the key: add a positive distance. Memory cue: right means plus and left means minus; then use a common denominator.

Correct answer: A. (8)

For ( -\sqrt{10}<-\sqrt{m}), we need ( \sqrt{10}>\sqrt{m}), so (m<10). Inequality reverses with negative roots.

Correct answer: A. ( \frac{5}{3},1.7,\sqrt{3})

( \frac{5}{3}\approx1.667), (1.7=1.7), and ( \sqrt{3}\approx1.732). Convert mixed forms to decimals for ordering.

Correct answer: A. (-7) and (-6)

The point has coordinate \\( -\sqrt{48} \\). To locate it, compare 48 with nearby perfect squares: \\(6^2=36\\) and \\(7^2=49\\). Therefore \\(6<\sqrt{48}<7\\). Taking the negative changes the order on the number line, so \\(-7<-\sqrt{48}<-6\\). Thus the point lies between the consecutive integers \\(-7\\) and \\(-6\\), which is option A. A common mistake is to choose 6 and 7 because the square root itself lies there, without considering the negative sign.

The value is approximately \\(-6.93\\), confirming the interval. The negative of a number between 6 and 7 must lie between -7 and -6, not between -6 and -5. The endpoint values are not included because \\(48\\) is not a perfect square, so the inequalities are strict. Hence the supplied answer A follows directly from comparing perfect squares and reversing the order correctly for negative numbers.

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

( \sqrt{5}\approx2.236), so (2-\sqrt{5}\approx-0.236). Estimation is fastest for subtraction with roots.

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

( \sqrt{\frac{25}{16}}=\frac{5}{4}) because the principal square root is positive. Take the root of both numerator and denominator.

Correct answer: A. (2.5)

( \sqrt{6}\approx2.449) and ( \sqrt{7}\approx2.646), so (2.5) lies between them. Estimate carefully for close square roots.

Correct answer: A. (1.25)

( \sqrt{1.44}=1.2) and ( \sqrt{1.69}=1.3), so (1.25) lies between them. Knowing decimal squares is useful.

Correct answer: A. ( \pi,\frac{19}{6},\sqrt{10})

( \pi\approx3.14159), ( \frac{19}{6}\approx3.1667), and ( \sqrt{10}\approx3.1623), so exact checking is needed. Estimate each value accurately while comparing.

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

Since ( \sqrt{5}\approx2.236), ( \sqrt{5}+\frac{1}{2}\approx2.736). Use estimation to identify the interval quickly.

Correct answer: A. (3.8)

( \sqrt{14}\approx3.742), so (3.8) lies between it and (4). Decimal estimation is important for nearby square roots.

Correct answer: A. ( -\frac{11}{3})

( -\frac{11}{3}\approx-3.667) and ( -\sqrt{13}\approx-3.606), so ( -\frac{11}{3}) is smaller. On a number line, the smaller number lies farther left.

Correct answer: B. 7 and 8

Compare the nearest perfect squares: \(7^2=49\) and \(8^2=64\). Since \(49<52<64\), taking square roots gives \(7<\sqrt{52}<8\). Therefore, \(\sqrt{52}\) lies between 7 and 8. Exam tip: To bound a square root, identify the consecutive perfect squares immediately below and above the number.

Correct answer: B. 2.875

Convert the fraction to a decimal by dividing 23 by 8: \(23 \div 8 = 2.875\). Therefore, \(\frac{23}{8}\) corresponds to the point 2.875 on the number line. Options A and D represent \(\frac{21}{8}\) and \(\frac{19}{8}\), respectively, so they are incorrect. Exam tip: Convert a fraction to a decimal or mixed number before locating it on the number line.

Correct answer: C. ( -3+\sqrt{14} )

Moving right means adding the distance, so the coordinate is ( -3+\sqrt{14} ). Choose addition or subtraction by direction.

Correct answer: B. \(1.8\)

Since \(1.8^2=1.8\times1.8=3.24\), we have \(\sqrt{3.24}=1.8\). The square root symbol denotes the positive square root, so the corresponding point has coordinate \(1.8\). The closest distractor, \(1.6\), is incorrect because \(1.6^2=2.56\). Exam tip: verify a decimal square root by squaring the proposed value and checking its place value.

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

( -\frac{19}{6}\approx-3.167 ), so it lies between (-4) and (-3). Converting a negative fraction to decimal is useful.

Correct answer: A. \(a<c<b\)

Since \(\sqrt{6}\approx2.449\), \(c=2.48\), and \(\frac{5}{2}=2.5\), we get \(2.449<2.48<2.5\). Therefore, the increasing order is \(a<c<b\). Remember that increasing order means arranging numbers from least to greatest.

Correct answer: B. 8.1

Since \(8^2=64\) and \(9^2=81\), \(\sqrt{65}\) is very close to 8. More precisely, \(\sqrt{65}\approx8.062\), which rounds to 8.1 to one decimal place. Option 7.9 lies below 8 and is not the correct nearest decimal estimate. Exam tip: bracket a square root between two consecutive perfect squares, then round its decimal value to the required place.

Correct answer: C. (4.65)

( \sqrt{21}\approx4.583 ) and ( \sqrt{22}\approx4.690 ), so (4.65) lies between them. Estimate accurately for close roots.