Representing real numbers on the number line

The governing number-line rule is that the greater real number lies farther to the right. Convert the fraction to a decimal: 3/2 = 1.5. Comparing 1.5 and 1.6 gives 1.6 > 1.5, so 1.6 has the greater value and must be located to the right of 3/2. Hence option B is correct. Option A would be correct only if 1.5 were greater than 1.6, which it is not. Option C is incorrect because the values differ, and option D is incorrect because 1.6 is greater than 1, while 3/2 is also 1.5, so neither number is less than 1. Converting both representations to a common form avoids confusion.

The governing concept is directed movement on a number line. Moving to the right increases a coordinate, while moving to the left decreases it. Starting at 2 and moving 1.5 units left therefore requires subtraction: Q = 2 − 1.5 = 0.5. Thus option B is correct. A quick check is that 0.5 is indeed 1.5 units to the left of 2, because 2 − 0.5 = 1.5. Option A comes from adding instead of subtracting, option C gives the distance moved rather than the final coordinate, and option D would be obtained by moving 2.5 units left or by using an incorrect subtraction. The direction word “left” determines the operation.

On a number line, the length between two points is the distance separating their coordinates. Distance cannot be negative, so we use the absolute value of the difference. If one point is at 1.4 and the other is at 2.1, the second point lies to the right of the first. The segment length is therefore found by subtracting the smaller coordinate from the larger one.

Using the distance rule, write AB = \(|2.1 - 1.4|\). The subtraction gives 0.7, and its absolute value is also 0.7. Thus the points are seven-tenths of one unit apart, so option B is correct. The value 1.7 would come from adding the coordinates, which does not measure the distance between points on a number line.

The governing concept is the midpoint property: the midpoint of two numbers a and b is their arithmetic mean, (a + b)/2. Here the endpoints are −2 and 4, so M = (−2 + 4)/2 = 2/2 = 1. Geometrically, 1 is equally far from both endpoints: its distance from −2 is 3 units and its distance from 4 is also 3 units. Therefore option A is correct. Option B results from using only half of the positive endpoint, option C reflects an incorrect sign calculation, and option D is not equidistant from the two given points. Adding signed numbers carefully is important: −2 + 4 equals 2, not −6 or 6.

The governing concept is the order of real numbers according to their signs. Every negative number lies to the left of zero, while every positive number lies to the right. Since √2 is positive, its opposite, −√2, is negative and therefore lies to the left of 0. Hence option C is correct. Option A reverses the position of a negative number, and option B incorrectly places the positive number √2 on the left. Option D is also false: √2 is irrational and approximately 1.414, not exactly 2. In fact, if √2 were 2, squaring both sides would give 2=4, an impossibility. The sign of −√2 alone is sufficient to determine its position on the number line.

Answer: C, 17/8. Equal numbers represent the same point on a number line, so convert 2.125 exactly into a fraction. Since 2.125 has three decimal places, 2.125=2125/1000. Dividing numerator and denominator by 125 gives 17/8. A quicker method is 2.125=2+0.125=2+1/8=16/8+1/8=17/8. Therefore C is the only equivalent fraction. A is 15/8=1.875, so it marks a smaller point. B is 16/8=2, not 2.125. D is 19/8=2.375, so it marks a larger point. Do not compare only the numerators; always reduce or convert the fractions to their values. The exact equality, not approximate closeness, determines the same point.

The governing concept is equal partition of a distance on the number line. The total distance between the points 2 and 5 is the absolute difference |5-2|=3 units. Dividing this total length into 6 equal parts gives 3/6=1/2 unit for each part. Hence option B is correct. Option A would result from dividing 3 by 9, option C would incorrectly use a unit length, and option D is the entire distance rather than one of six equal portions. Using the absolute difference is important because distance is always non-negative, regardless of the direction in which the points are named.

The governing idea is repeated equal movement along a number line. Starting at 2, each step has length 1/2. After four steps, the total movement is 4×1/2=2 units. Adding this to the starting position gives 2+2=4, so option C is correct. Equivalently, the successive points are 2.5 after one step, 3 after two, 3.5 after three, and 4 after four. Option A stops after two steps, option B after three steps, and option D moves five steps. The wording “fourth point” is interpreted as four equal intervals from the starting point, consistent with the preceding division setup.

The governing concept is simplification of a square root before performing division. Because 16 is a perfect square, its principal square root is √16 = 4, not ±4 in this context. Hence √16/2 = 4/2 = 2. The corresponding point on the number line is therefore 2, so option A is correct. Option B results from stopping after evaluating the square root and forgetting the division by 2. Option C incorrectly multiplies 4 by 2, while option D reverses the division and treats the expression as 2/4. The expression is positive, so no negative interpretation is involved.

The governing concept is the midpoint formula for two points on a number line. If the endpoints are a and b, their midpoint is M=(a+b)/2, because this average is equally distant from both endpoints. Substituting a=-1.5 and b=4.5 gives M=(-1.5+4.5)/2=3/2=1.5. The result can also be checked geometrically: the distance from -1.5 to 1.5 is 3 units, and the distance from 1.5 to 4.5 is also 3 units. Therefore option A is correct. Option B is the total of the two values after an incorrect operation, option C is not their average, and option D does not divide the interval into two equal parts.

Movement to the left on a number line means subtracting the stated distance from the starting coordinate. The starting point is 7/10 and the movement is 3/20 unit, so calculate 7/10 − 3/20. Convert 7/10 to twentieths: 7/10 = 14/20. Then 14/20 − 3/20 = 11/20. Therefore the required point is 11/20, making option A correct. Option B adds the distance and therefore represents movement to the right. Option C is 10/20 and does not result from the required subtraction, while option D uses an incorrect denominator conversion. The direction word “left” determines the operation.

On a number line, the distance of a point x from zero is |x|. The equation |x| = √17 has two solutions: x = √17 and x = −√17. The additional condition says that the point must lie to the left of zero. Every point left of zero has a negative coordinate, so the required solution is −√17. Therefore option A is correct. Option B has the same distance from zero but lies to the right. Option C has distance 17, not √17, and option D has distance 17 as well, despite being negative. The direction condition is essential for choosing between the two points.

The governing concept is simplifying a real-number expression before locating its value on the number line. The principal square root of 4 is √4 = 2, because 2² = 4 and the principal root is non-negative. Therefore, 1/√4 = 1/2. This point is positive and lies halfway between 0 and 1, so option A is correct. Option B is the denominator after simplification, not its reciprocal. Option C would result from incorrectly replacing √4 by 4 and then taking 1/4. Option D is the radicand itself, not the value of the given expression. The exact simplification leaves no need for approximation.

The governing concept is converting a rational number from fractional form to decimal form while preserving its sign. Dividing 7 by 8 gives 7 ÷ 8 = 0.875; equivalently, 7/8 can be written as 875/1000. Since the original fraction is negative, -7/8 = -0.875. Therefore option A is correct. Option B changes the order of the digits and has a different value. Option C has the correct magnitude but loses the negative sign, so it represents positive 7/8 and lies to the right of zero. Option D is also negative but its magnitude is 1.125, which is not equal to 0.875. Exact fraction-to-decimal conversion confirms the answer.

The governing concept is the midpoint formula: the point exactly halfway between two real numbers is their arithmetic mean. For the endpoints 5/12 and 7/12, calculate M = [(5/12) + (7/12)]/2. Since the denominators are equal, the sum is 12/12 = 1; dividing by 2 gives M = 1/2. Therefore option B is correct. A distance check gives the same result: 1/2 = 6/12, so it is one twelfth to the right of 5/12 and one twelfth to the left of 7/12. Option A, 1/3 = 4/12, lies to the left of the first endpoint. Option C, 2/3 = 8/12, lies to the right of the second endpoint, and option D is farther right. Hence only B is exactly midway.

The governing concept is the relationship between distance from zero and absolute value on a number line. A point whose distance from 0 is √26 must have absolute value √26, so the two algebraic possibilities are √26 and -√26. The additional condition says that the point lies to the right of 0, and every point to the right of zero has a positive coordinate. Therefore the required number is √26, making option B correct. Option A has the correct distance because |-√26| = √26, but it lies to the left. Options C and D have absolute value 26, not √26, so their distances from zero are different. Direction resolves the otherwise two-valued distance condition.

The governing concept is directed distance on the number line. A point whose distance from zero is √41 can have either coordinate √41 or −√41, since equal distances occur on opposite sides of zero. The additional condition “to the left of 0” selects the negative coordinate, so the answer is −√41. Option B is correct. Option A has the required distance but lies to the right of zero. Options C and D are not correct because the distance of either 41 or −41 from zero is 41, not √41. Thus both magnitude and direction must be used. This distinction is important: ordinary distance is nonnegative, while the coordinate records the side of zero through its sign.

The governing concept is evaluating exact square roots and then performing the indicated subtraction. Since 256 = 16², √256 = 16. Since 121 = 11², √121 = 11. Therefore the coordinate of P is √256 − √121 = 16 − 11 = 5, so option A is correct. No decimal approximation is needed because both radicands are perfect squares. Option B comes from adding the two roots instead of subtracting them. Option C has no valid basis in the calculation, while option D is only √256 and ignores the second term. The coordinate is therefore an exact real number, and its positive value places P five units to the right of zero on the number line.