Representing real numbers on the number line

Given \(\sqrt{a}=4.5\), squaring both sides gives \(a=(4.5)^2=20.25\). Therefore, option C is correct. Option A is only twice \(4.5\), but removing a square root requires squaring the number. Exam tip: whenever \(\sqrt{a}=x\), write \(a=x^2\).

First, \(\sqrt{0.04}=0.2\). Moving \(0.2\) units to the right on the number line means adding \(0.2\) to \(-0.4\): \(P=-0.4+0.2=-0.2\). Option A results from subtracting \(0.2\), which represents moving left instead. Exam tip: moving right increases the number, while moving left decreases it.

(\sqrt{6}\approx2.45), so (-\sqrt{6}\approx-2.45), which is left of (-2). The smaller number lies to the left.

The direct answer is option B: \(\sqrt{1},\sqrt{2},\sqrt{3}\). For non-negative numbers, the square-root function preserves order: if \(1<2<3\), then \(\sqrt{1}<\sqrt{2}<\sqrt{3}\). We may also evaluate them: \(\sqrt{1}=1\), \(\sqrt{2}\approx1.414\), and \(\sqrt{3}\approx1.732\). Therefore they appear from left to right in the stated order. Option A reverses the order, so it is decreasing rather than increasing. Option B follows the increasing order and is correct. Option C places \(\sqrt{2}\) before \(\sqrt{1}\), which is impossible because \(\sqrt{2}>1\). Option D puts \(\sqrt{3}\) before \(\sqrt{2}\), although \(\sqrt{3}>\sqrt{2}\). The useful rule is to compare the non-negative numbers inside the roots; larger radicand gives larger square root.

The direct answer is option B, 1. First simplify the endpoints: \(-\sqrt{4}=-2\), because the negative sign is outside the principal square root, and \(\sqrt{16}=4\). The midpoint of two numbers \(a\) and \(b\) is \(\frac{a+b}{2}\). Hence \(\frac{-2+4}{2}=\frac{2}{2}=1\). Option A, 0, would be the midpoint of -2 and 2, not -2 and 4. Option B, 1, is exactly halfway: it is 3 units from -2 and 3 units from 4. Option C, 2, is closer to 4 and is not halfway. Option D, 3, is also closer to 4. A common error is to treat \(-\sqrt{4}\) as positive; the minus sign must remain, so the endpoint is -2. Exam cue: simplify first, then average the two coordinates.

(\frac{13}{4}=3+\frac{1}{4}), so it is the first quarter point after (3). Think of improper fractions as mixed numbers.

(\sqrt{5}\approx2.24), which is closest to (2) among the options. Compare each distance for closeness.

(-\sqrt{25}=-5) and (-\sqrt{36}=-6), so (-5) is to the right. Among negatives, the one with smaller magnitude is greater.

The direct answer is option B, 0.9. First calculate the square root: \(\sqrt{0.81}=0.9\), since \(0.9^2=0.81\). The point is \(-\sqrt{0.81}=-0.9\). Distance from zero is the absolute value, so \(|-0.9|=0.9\). Option A, 0.09, is incorrect because \(0.09^2=0.0081\), not 0.81. Option B, 0.9, is correct. Option C, 1.8, doubles the distance and would represent the distance between -0.9 and +0.9, not between -0.9 and zero. Option D, 9, ignores the decimal place and is much too large. Notice that the negative sign tells the side of zero, while distance tells how far and is non-negative. Memory cue: position may be negative, but distance from zero is its magnitude.

Since (4^2<24<5^2), (\sqrt{24}) is between (4) and (5) and is not equal to (24). Do not treat a square root as the original number.

For the symmetric point (3) is the midpoint so the other point is (2\times3-4.6=1.4). In symmetry the midpoint is equally distant from both endpoints.

Since (6^2<37<7^2), (6<\sqrt{37}<7) and (0<\sqrt{37}-6<1). Subtract the same number from a root interval to locate the value.

The governing property is that for a non-negative number a, if 0<a<1, then its principal square root also satisfies 0<\sqrt{a}<1. Here 0<0.49<1, so taking the non-negative square root gives 0<\sqrt{0.49}<1. In fact, \sqrt{0.49}=0.7, which directly confirms the statement. Therefore option A is correct. Option B contradicts the given value, option C confuses the square root with the number 49, and option D is impossible because a principal square root is never negative. The interval property, rather than an unsupported approximation, provides the justification.

The governing concept is bounding an irrational expression between consecutive integers. Because 1² < 2 < 2², taking positive square roots gives 1 < √2 < 2. Adding 3 to every part preserves the order, so 4 < 3 + √2 < 5. Thus the expression lies strictly between 4 and 5, making option A correct. A decimal check gives √2 ≈ 1.414 and therefore 3+√2 ≈ 4.414, which confirms the interval. Option B is incorrect because the value is greater than 4. Option C is incorrect because the value is less than 5, and option D places it even farther below the actual value. The exact inequality is sufficient; decimal approximation is only a confirmation.

The governing concept is order preservation when squaring positive numbers. Since both 2.6 and 2.7 are positive, the statement 2.6 < √7 < 2.7 is equivalent to 2.6² < 7 < 2.7². Calculate the endpoint squares: 2.6² = 6.76 and 2.7² = 7.29. Because 6.76 < 7 < 7.29, the required test is true, and √7 does lie in that interval; numerically it is about 2.646. Therefore option A is correct. Option B reverses the natural order of the endpoint squares. Option C says 7 is below the lower square, and option D says it is above the upper square; both contradict the actual inequalities and fail to establish that √7 lies between the given bounds.

The governing concept is that the distance between two points on a number line is the absolute value of the difference of their coordinates. Therefore AB=|1.2−(−3.8)|=|5.0|=5 units. Similarly, AC=|1.2−4.2|=|−3.0|=3 units. The question asks how much greater AB is than AC, so subtract the two distances: 5−3=2 units. Hence option A is correct. Absolute value is essential because distance cannot be negative, even when a directed difference is negative. Option B does not equal the difference of the calculated distances, while options C and D are not supported by the coordinate calculation.

The governing concept is comparing rational and irrational numbers by using exact values or sufficiently accurate decimal approximations. First, 13/4=3.25 exactly. Also, √11 is approximately 3.3166; this is reasonable because 3.3166² is close to 11 and √11 lies between √9=3 and √16=4. The three values are therefore 3.25, 3.28, and approximately 3.3166. Since 3.25<3.28<3.3166, the increasing order is 13/4, 3.28, √11. Hence option A is correct. Option B reverses the first two values, C makes √11 the smallest, and D places √11 before 3.28, both contrary to the approximations.

The key concept is comparing negative real numbers in their actual number-line order. Since √3≈1.732, we have -√3≈-1.732. Also, -5/3=-1.6666... . A number between these endpoints must be greater than -1.732 and less than -1.6666..., because the more negative number is farther to the left. Among the choices, -1.70 satisfies -1.732<-1.70<-1.6666..., so it lies between the two given values. Therefore option A is correct. The value -1.60 is to the right of the interval, while -1.80 and -2.00 are more negative and lie to its left. Treating the negative values as positive magnitudes would reverse the comparison incorrectly.

The governing concept is estimating an irrational square root by locating it between consecutive perfect squares and comparing decimal approximations. Since 7² = 49 and 8² = 64, √63 lies between 7 and 8 and must be very close to 8. Testing the nearest plausible option, 7.94² = 63.0436, whose difference from 63 is only 0.0436. More accurately, √63 is approximately 7.937253, which rounds to 7.94. Therefore option A is the closest decimal value. Option B, 7.50, is substantially below the root; option C, 8.20, is above 8 and therefore farther away; and option D, 6.90, is even outside the interval from 7 to 8. Thus the number-line position confirms A.

The governing concept is comparison of negative real numbers within an inequality. First, √50 = 5√2 ≈ 7.071, so -√50 ≈ -7.071. The condition therefore requires x to be greater than -7.071 but less than -7; in interval notation, x must belong to approximately (-7.071, -7). The value -7.05 lies inside this narrow interval, so option A is possible. The value -7.20 is less than -7.071 and lies outside on the left. The value -6.90 is greater than -7, and -8.00 is also smaller than the left endpoint. Because negative numbers increase when they move rightward on the number line, careful order checking is essential.

The governing concept is bounding a square root between consecutive integers and then applying the effect of a negative sign. Since 25 < 29 < 36, taking principal square roots gives 5 < √29 < 6. Multiplying all parts by -1 reverses the inequality signs, so -6 < -√29 < -5. Therefore x lies between -6 and -5, making option B correct. Options C and D describe positive intervals and ignore the negative sign in x. Option A is also incorrect because -√29 is less than -5 but greater than -6; it is not between -5 and -4. The bounds are exact enough to identify the interval without calculating a decimal approximation.

The governing concept is the midpoint formula for two points on a number line: M = (A + B)/2. Substitute the given values carefully: M = [(-13/2) + (-3/2)]/2 = (-16/2)/2 = (-8)/2 = -4. Therefore option B is correct. A geometric check gives the same result: A = -6.5 and B = -1.5, so -4 is 2.5 units from each endpoint. Option A may arise from adding the numerators but forgetting the final division by 2. Option C results from an incorrect arithmetic operation, while option D does not represent the average of the two endpoints. The exact formula establishes a unique midpoint.

The governing concept is ordering real numbers written in different forms by converting them to comparable values. First, 7/4 = 1.75. Also, √3 is approximately 1.732, while the third number is already 1.76. Thus the values compare as 1.732 < 1.75 < 1.76. Consequently, the correct increasing order is √3, 7/4, 1.76, which is option A. Option B reverses the first two values. Option C places the largest value first, and option D incorrectly places 1.76 before 7/4. The approximations are sufficiently separated, so no closer calculation is needed; alternatively, √3 < 1.75 can be confirmed by squaring positive quantities.

The governing concept is locating a negative irrational number by first bounding its positive square root between consecutive integers. Since 64 < 80 < 81, taking square roots gives 8 < √80 < 9. When each part of this inequality is multiplied by -1, the inequality signs reverse, producing -9 < -√80 < -8. Therefore P lies strictly between -9 and -8, so option A is correct. Option C gives the interval for the positive number √80 rather than P. Option B places P too close to zero, while option D places it below -9. The sign reversal is essential because negative numbers are ordered oppositely when multiplied by -1.

The governing concept is the order of negative real numbers and the reversal caused when a positive square root is multiplied by −1. Since 9²=81 and 10²=100, we know that 9<√82<10. A closer estimate is √82≈9.055, so −√82≈−9.055. Therefore x must lie strictly between −9.055 and −9. The proposed value −9.03 lies in this interval because −9.055<−9.03<−9, so it satisfies both strict inequalities. The value −9.20 is smaller than −9.055, −8.90 is greater than −9, and −10 is also below the permitted interval. Hence option A is the only possible value.