Representing real numbers on the number line

The midpoint of two numbers is found using (\frac{a+b}{2}). Here, (\frac{-4+2}{2}=\frac{-2}{2}=-1), so the correct answer is (-1). Choosing (1) means the negative sign was ignored after addition. Exam tip: Find the average of the coordinates of the two points to get their midpoint.

Moving to the right on a number line means adding units. Therefore, moving 5 units right from 0 gives 0 + 5 = 5. The point -5 would be reached by moving 5 units to the left. Exam tip: rightward movement represents positive change, while leftward movement represents negative change.

Moving 6 units to the left from 0 decreases the value by 6, so the point reached is 0 − 6 = −6. Option B, 6, would be reached by moving 6 units to the right. Exam tip: on a number line, the left direction represents negative movement and the right direction represents positive movement.

On a number line, moving to the right means adding the distance travelled. Therefore, moving 3 units right from 2 gives 2 + 3 = 5. The point -1 would be reached by moving 3 units left, while 3 is only the distance travelled, not the final point. Exam tip: add when moving right and subtract when moving left.

Moving left on the number line decreases the value. Therefore, moving 5 units left from 2 gives 2 - 5 = -3. Option B results from adding instead of subtracting. Exam tip: subtract for a leftward movement and add for a rightward movement.

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.

( \frac{5}{2}=2.5), so it lies between (2) and (3). Converting a fraction to decimal is an easy method.

The governing concept is ordering negative rational numbers on the number line. Convert the fraction to a decimal: −5/2 = −2.5. Now compare it with the neighboring integers: −3 < −2.5 < −2. Therefore, the point representing −5/2 lies between −3 and −2. The negative sign must be retained; ignoring it could incorrectly suggest the interval from 2 to 3. Option B is wrong because −2.5 is less than −2, not greater than −2. Options C and D contain positive numbers and therefore cannot contain a negative number. Hence option A is the only interval satisfying the 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.

Since (0.8>0.6), (0.8) lies to the right. The greater number is always on the right on a number line.

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.

Since (2^2<8<3^2), (\sqrt{8}) lies between (2) and (3). First identify nearby perfect squares.

The governing method is comparison with consecutive perfect squares. Since 3² = 9 and 4² = 16, and 13 lies between 9 and 16, we have 9 < 13 < 16. The square-root function is increasing for nonnegative numbers, so taking principal square roots gives 3 < √13 < 4. Therefore √13 lies between the integers 3 and 4, making option A correct. It cannot lie between 2 and 3 because that interval corresponds to squares between 4 and 9, and it cannot lie between 4 and 5 because it is less than 4. An approximation, √13 ≈ 3.606, confirms the result but is not required.

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 3 units away from 2 when the distance equation |x − 2| = 3 holds. Removing the absolute value gives two cases: x − 2 = 3, which yields x = 5, and x − 2 = −3, which yields x = −1. Geometrically, moving three units right from 2 reaches 5, while moving three units left reaches −1. Thus the required pair is −1 and 5, so option A is correct. Option B contains points one unit from 2. In option C, −3 is five units from 2, and option D gives distances two and one, so neither satisfies the condition.

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.