Representing real numbers on the number line

Correct answer: A. 5

Moving to the right on a number line means adding a positive number. Thus, −2 + 7 = 5, so the final point is 5. The option −9 incorrectly keeps a negative sign after adding the magnitudes, while 7 ignores the starting point. Exam tip: moving right increases the number, whereas moving left decreases it.

Correct answer: A. -(7) / (-7)

On a number line, moving to the left means moving in the negative direction. Each unit of leftward movement decreases the number by one. Starting at \\(-3\\) and moving four units left therefore means subtracting \\(4\\) from \\(-3\\), not adding it. This gives a point farther from zero on the negative side.

The calculation is \\(-3-4=-7\\). The four successive positions are \\(-4,-5,-6,-7\\), so the final point is \\(-7\\). Hence option A is correct. A common error is to use \\(-3+4=1\\), but adding four represents movement to the right, not to the left. The direction on the number line determines the sign of the change.

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

The direct answer is A: \(\frac52\) lies between 2 and 3. A fraction \(\frac52\) means 5 divided by 2. Dividing gives \(2.5\). Since \(2<2.5<3\), its point on the number line is between the consecutive integers 2 and 3. Option A is correct. Option B, 1 and 2, cannot contain 2.5 because 2.5 is greater than 2. Option C, 3 and 4, is too far to the right because 2.5 is less than 3. Option D, 0 and 1, is also too small. Another method is to write \(\frac52=2+\frac12\): start at 2 and move halfway toward 3. The exam cue is to compare the fraction with nearby integers, or convert it to a decimal carefully.

Correct answer: A. −3 and −2

The governing concept is ordering rational numbers on a number line. Convert the fraction to a decimal: -5/2=-2.5. Now compare it with consecutive integers: -3<-2.5<-2. Therefore the point representing -5/2 lies between -3 and -2, so option A is correct. The negative sign is essential; removing it would incorrectly produce the interval from 2 to 3. Option B is wrong because -2.5 is less than -2, so it cannot lie between -2 and -1. Options C and D contain only positive numbers, whereas -5/2 is negative. The same conclusion follows by multiplying the inequalities by 2: -6<-5<-4, which confirms -3<-5/2<-2. Thus A is the only interval satisfying both required inequalities.

Correct answer: A. (10) parts

(0.4=\frac{4}{10}), so it is useful to divide (0) to (1) into (10) equal parts. Then move (4) parts ahead.

Correct answer: A. (0.8)

The direct answer is option A: \(0.8\) lies to the right. On a number line, numbers increase as we move from left to right. Compare the decimal parts: both numbers have zero units, and \(8\) tenths is greater than \(6\) tenths, so \(0.8>0.6\). Therefore, 0.8 is farther right. Option A is correct. Option B, 0.6, is the smaller number and lies left of 0.8. Option C, 0.0, is less than both numbers, so it lies farther left. Option D, \(-0.8\), is negative and is also left of both positive numbers. A useful comparison is to write \(0.6=6/10\) and \(0.8=8/10\); since 8 tenths is greater than 6 tenths, the position is rightward. Memory cue: on a number line, greater means right and smaller means left.

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

Writing 3.5 as a fraction gives \(3.5=\frac{35}{10}=\frac{7}{2}\), so option A is correct. The other options have different values; for example, \(\frac{3}{5}=0.6\), not 3.5. Exam tip: when a decimal has one digit after the decimal point, write it over 10 and simplify.

Correct answer: A. Legs (1) and (1)

If the legs are (1) and (1), the hypotenuse is (\sqrt{1^2+1^2}=\sqrt{2}). This length is transferred to the number line with an arc.

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

In a right triangle with legs (1) and (2), the hypotenuse is (\sqrt{1^2+2^2}=\sqrt{5}). Such lengths come from the Pythagoras theorem.

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

The direct answer is option A: \(\sqrt8\) lies between 2 and 3. To locate a square root, compare nearby perfect squares. Since \(2^2=4\) and \(3^2=9\), we have \(4<8<9\). Taking positive square roots gives \(2<\sqrt8<3\). Therefore, option A is correct. Option B, between 1 and 2, is wrong because numbers in that interval have squares between 1 and 4, while 8 is larger than 4. Option C, between 3 and 4, is wrong because \(\sqrt8<\sqrt9=3\). Option D, between 4 and 5, is even farther right and cannot work. Although \(\sqrt8\) is approximately 2.83, approximation is not necessary. The key is identifying the consecutive perfect squares surrounding 8. Memory cue: square the endpoint integers before deciding.

Correct answer: A. 3 and 4

The governing method is comparison with consecutive perfect squares. Since 3^2=9 and 4^2=16, and 9<13<16, taking principal square roots gives 3<√13<4. Hence √13 lies between the integers 3 and 4, so option A is correct. It cannot lie between 2 and 3 because numbers in that interval have squares between 4 and 9, whereas 13 is larger than 9. It cannot lie between 4 and 5 because √13 is less than 4, as 13 is less than 16. A decimal approximation, √13≈3.606, confirms the location but is unnecessary. The comparison method is exact and avoids relying on rounded values or an inaccurate estimate.

Correct answer: A. 1.45

Write the numbers with the same number of decimal places: 1.4 = 1.40 and 1.5 = 1.50. Since 1.40 < 1.45 < 1.50, 1.45 lies between them. The distractor 1.35 is less than 1.4, while 1.55 is greater than 1.5. Exam tip: add trailing zeros to decimals before comparing their place values.

Correct answer: A. −1 and 5

The governing concept is absolute distance on a number line. A point x is three units away from 2 exactly when |x-2|=3. This absolute-value equation has two cases: x-2=3, giving x=5, and x-2=-3, giving x=-1. Geometrically, moving three units to the right from 2 reaches 5, while moving three units to the left reaches -1. Therefore the required points are -1 and 5, so option A is correct. Option B contains 1 and 3, each only one unit from 2. In option C, -3 is five units from 2 and 2 is zero units away. In option D, 0 is two units away and 3 is one unit away. Thus only A satisfies the distance condition for both points.

Correct answer: A. −6 and 2

The governing concept is absolute distance on a number line. A point x is 4 units away from −2 when the distance between x and −2 is 4, represented by |x − (−2)| = 4, or |x + 2| = 4. An absolute-value equation has two cases: x + 2 = 4, which gives x = 2, and x + 2 = −4, which gives x = −6. Therefore, the two points are −6 and 2: one is four units to the left of −2 and the other is four units to its right. Option B gives distances 2 and 6, option C includes the original point −2, and option D gives distances 2 and 8. Hence option A is correct.

Correct answer: A. a is to the right of b

The governing concept is the geometric interpretation of inequality on a number line. Numbers increase as we move from left to right. Therefore, when a > b, the value of a is greater than the value of b, so the point representing a must lie to the right of the point representing b. For example, if a = 5 and b = 2, the point 5 is to the right of 2. The same idea works for negative numbers: −1 > −4, and −1 is also to the right of −4. Option B reverses the order, option C contradicts the strict inequality by claiming equality, and option D states the opposite spatial relation. Thus option A is the only correct answer.

Correct answer: A. 1

The midpoint of two numbers is found using \(\frac{a+b}{2}\). Here, \(\frac{0+2}{2}=1\), so the correct answer is 1. \(\frac{1}{2}\) is the midpoint of 0 and 1, not of 0 and 2. Exam tip: calculate the average of the two endpoints to find their midpoint.

Correct answer: A. 0

The midpoint of two numbers is their average: (−1 + 1) ÷ 2 = 0. Therefore, 0 lies exactly halfway between −1 and 1.

Correct answer: A. (\frac{9}{4})

A decimal can be changed into a fraction by writing it over a power of 10. The number 2.25 has two digits after the decimal point, so 2.25=225/100. Now reduce this fraction by dividing numerator and denominator by their common factor 25: 225÷25=9 and 100÷25=4. Thus 2.25=9/4, making choice A correct.

The result can also be checked by dividing 9 by 4: 4 goes into 9 twice with remainder 1, and the remainder gives 0.25, so the quotient is 2.25. The fraction 4/9 is less than 1, 5/4 equals 1.25, and 11/4 equals 2.75, so none of those represents 2.25. Therefore the point on the number line has value 9/4.

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

The middle number between (0) and (1) is (\frac{0+1}{2}=\frac{1}{2}). In exams, use the average for the midpoint.

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

(-\frac{3}{2}=-1.5), so it lies between (-2) and (-1). For negative numbers, values increase to the right.

Correct answer: A. (4)

The denominator of (\frac{3}{4}) is (4), so divide (0) to (1) into (4) equal parts. The numerator tells you to move to the third mark.

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

(-\frac{7}{3}\approx -2.33), so it lies between (-3) and (-2). Check the position of negative decimals carefully.

Correct answer: A. 5/4

A terminating decimal can be converted into a fraction by writing it over the appropriate power of ten. Since 1.25 has two digits after the decimal point, 1.25 = 125/100. Dividing the numerator and denominator by their common factor 25 gives 125/100 = 5/4. Therefore the point representing 1.25 on the number line is the point 5/4 units to the right of zero. Option A is correct. The value 4/5 equals 0.8, so option B lies to the left of 1; 3/4 equals 0.75, so option C is also smaller; and 7/4 equals 1.75, so option D lies farther to the right. The governing idea is equivalence between decimal and fractional representations, not merely matching the digits 1 and 25.

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

Converting 0.6 into a fraction gives \(0.6=\frac{6}{10}=\frac{3}{5}\). Therefore, 0.6 and \(\frac{3}{5}\) represent the same point on the number line. Option B, \(\frac{5}{3}\), is greater than 1, whereas 0.6 is less than 1. Exam tip: write the decimal over 10, 100, or 1000 according to its place value, then simplify.

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

The integer part of (2.75) is (2), so it lies between (2) and (3). The integer part gives the first clue.