Representing real numbers on the number line

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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

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

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.

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

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.

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.

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