Representing real numbers on the number line

Correct answer: B. (6) parts

The denominator of (\frac{5}{6}) is (6), so the segment from (0) to (1) has (6) equal parts. In exams, the denominator tells the number of parts.

Correct answer: A. Third mark after (1) when there are (8) equal parts

(1+\frac{3}{8}) is the third of (8) equal parts after (1). In exams, count the fractional parts after the integer.

Correct answer: C. (4) units

The direct answer is option C, 4 units. Distance on a number line means the positive gap between two positions, so use the absolute difference: \(AB=\left|\frac52-\left(-\frac32\right)\right|\). Subtracting a negative becomes addition: \(\frac52+\frac32=\frac82=4\). Option A, 1 unit, is too small and does not equal the gap. Option B, 2 units, ignores the full movement across zero. Option C, 4 units, is correct; from \(-1.5\) to 0 is 1.5 units and from 0 to 2.5 is 2.5 units, totaling 4. Option D, 5 units, is an incorrect sum. The answer must be positive even though the first point is negative. Memory cue: distance equals right endpoint minus left endpoint, or always use absolute difference.

Correct answer: B. (4) units

The direct answer is option B, 4 units. Distance on a number line is the absolute difference between the two coordinates. Taking the larger coordinate minus the smaller one gives \(1.25-(-2.75)=1.25+2.75=4\). The negative sign changes subtraction into addition because subtracting a negative is adding. Option A, 3 units, is incorrect because it does not cover the whole interval from -2.75 to 1.25. Option B, 4 units, is correct. Option C, 4.5 units, results from an arithmetic error and is too large. Option D, 5 units, is also too large; it may come from rounding or adding incorrectly. A distance cannot be negative, and here the two points lie on opposite sides of zero, so their magnitudes add: 2.75+1.25=4. Exam cue: when one endpoint is negative and the other positive, add the distances from zero.

Correct answer: A. (-\frac{5}{6})

The midpoint is (\frac{-\frac{7}{3}+\frac{2}{3}}{2}=-\frac{5}{6}). In exams, first add and then divide by (2).

Correct answer: B. 4

The midpoint of two numbers is their average. Thus, \(\frac{2.4+5.6}{2}=\frac{8}{2}=4\), so option B is correct. Options A and C do not give the correct average, while 8 is the sum of the two numbers, not their midpoint. In an exam, add the two numbers and divide the result by 2 to find the midpoint.

Correct answer: B. (-2.5)

From (2x+5=0), we get (x=-\frac{5}{2}=-2.5). In exams, find the zero and place it like a real number.

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

From (3x-7=0), (x=\frac{7}{3}), which lies between (2) and (3). In exams, first find the zero and then locate it.

Correct answer: A. (-2) and (-1)

From (4x+6=0), (x=-\frac{3}{2}), which lies between (-2) and (-1). In exams, identify the interval of a negative fraction carefully.

Correct answer: B. Points \(-4\) and \(4\)

To find the zeros, set \(s(x)=0\): \(x^2-16=0\). Factoring gives \((x-4)(x+4)=0\), so \(x=4\) or \(x=-4\). Therefore, the corresponding points on the number line are \(-4\) and \(4\). Exam tip: for an equation of the form \(x^2=a^2\), check both values \(x=\pm a\).

Correct answer: D. (-\sqrt{7}) and (\sqrt{7})

From (x^2-7=0), (x=\pm\sqrt{7}). In exams, take both positive and negative square roots in a square equation.

Correct answer: A. (0) and (2)

(x^2-2x=x(x-2)), so the zeros are (0) and (2). In exams, taking a common factor is an easy method.

Correct answer: B. (2.4)

(\sqrt{6}) is about (2.45), so (2.4) is closest. In exams, think of (2.4^2) and (2.5^2) for estimation.

Correct answer: C. (3.4)

(\sqrt{11}) is about (3.32), so (3.4) is to its right. In exams, a point to the right is greater.

Correct answer: B. (3.5)

(\sqrt{14}) is about (3.74), so (3.5) is greater than (3) and less than (\sqrt{14}). In exams, make a rough estimate of the square root.

Correct answer: C. −2

The governing concept is that a point lies to the right of another point when its numerical value is greater. Since √5 is between 2 and 2.25 (because 2² = 4 and 2.25² = 5.0625), −√5 is approximately −2.236. Comparing the choices with −2.236, −2 is greater and therefore lies to its right. Options −3 and −4 are smaller, so they lie to the left. The value −2.5 is also smaller than approximately −2.236 and is therefore to the left. Thus option C is correct. With negative numbers, the number closer to zero is greater, which is the key point that prevents reversing the order.

Correct answer: A. (-\sqrt{3},-1,0,\sqrt{2})

Since (-\sqrt{3}) is about (-1.73), it is left of (-1). In exams, order increases from left to right.

Correct answer: A. 1/6 unit

The governing concept is that the distance between two points on a number line is the absolute difference of their coordinates. Here Q is to the right of P because 5/6 > 2/3. Use a common denominator: 2/3 = 4/6. Therefore the distance is |5/6 − 4/6| = |1/6| = 1/6 unit. Option A is correct. Option B would result from an incorrect denominator operation, option C treats the numerators or denominators separately without equivalence, and option D adds the two coordinates instead of finding their separation. Because the points are ordered, ordinary subtraction gives a positive distance; absolute value would also give the same result.

Correct answer: B. \(\frac{9}{10}\) unit

The distance is \(\frac{1}{10}-\left(-\frac{4}{5}\right)=\frac{9}{10}\) unit. In exams, subtracting a negative fraction becomes addition.

Correct answer: C. 2.55 units

The distance between two points on a number line is the absolute difference of their values. So, |3.75 − 1.20| = 2.55 units. Writing 1.2 as 1.20 aligns the decimal places for subtraction.

Correct answer: B. (2-\frac{3}{4})

Moving left decreases the number, so the point is (2-\frac{3}{4}). In exams, choose addition or subtraction according to direction.

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

The direct answer is option A, \(-\frac12\). On a number line, moving to the right means adding a positive distance. Starting at \(-3\) and moving \(\frac52\) units right gives \(-3+\frac52\). Write \(-3\) as \(-\frac62\), so \(-\frac62+\frac52=-\frac12\). Option A is therefore correct. Option B, \(-\frac{11}{2}\), results from moving left by \(\frac52\), because it is \(-3-\frac52\). Option C, \(\frac12\), would be obtained by adding \(\frac72\), not \(\frac52\), so it is too far to the right. Option D, \(-\frac52\), does not represent the stated movement; it is only one-half unit to the right of \(-3\). The sign direction is the key idea: right means addition and left means subtraction. A quick check is that moving 2.5 units right from -3 should leave the point still slightly negative, exactly \(-0.5\).

Correct answer: B. \(-1.5\)

Moving left on the number line means subtracting. Therefore, \(x=1-2.5=-1.5\). Option C, \(1.5\), does not result from subtracting \(2.5\) from \(1\). Exam tip: move right by adding and move left by subtracting.

Correct answer: C. (5) and (6)

Since (5^2=25) and (6^2=36), (\sqrt{27}) lies between (5) and (6). In exams, remember nearby perfect squares.

Correct answer: B. (-6) and (-5)

Since (\sqrt{30}) lies between (5) and (6), (-\sqrt{30}) lies between (-6) and (-5). In exams, keep the negative direction in mind.