Representing real numbers on the number line

Correct answer: A. 5√2

The governing concept is simplifying a square root by extracting perfect-square factors. Factor 50 as 25 × 2, where 25 is a perfect square. Using √(ab) = √a × √b, we get √50 = √(25 × 2) = √25 × √2 = 5√2. Therefore option A is correct. A numerical check gives √50 approximately 7.07, and 5√2 has the same approximate value. Although 2√5 is also approximately 4.47 and is not equal to √50, the other choices are much larger: 10√5 and 25√2 do not preserve the original value. The simplified radical should have no square factor remaining inside the root, which is why 5√2 is the preferred form.

Correct answer: B. (4) and (5)

(3\sqrt{2}) is about (4.24), so it lies between (4) and (5). In exams, you may estimate (\sqrt{2}) as about (1.414).

Correct answer: B. b

The governing concept is that the greater real number lies farther to the right on a number line. Compare the values without relying only on rounded decimals. Since 3.5 is positive, square it: 3.5²=12.25. Because 12<12.25 and both sides are non-negative, taking square roots preserves the order, so √12<√12.25=3.5. Numerically, √12 is approximately 3.464, which confirms the exact comparison. Therefore b=3.5 is to the right, making option B correct. Option A chooses the smaller value, option C is false because the values are unequal, and option D is unnecessary because the comparison is fully determined by squaring positive quantities.

Correct answer: B. (3) and (4)

(\frac{19}{6}=3+\frac{1}{6}), so it lies between (3) and (4). In exams, divide to identify the whole part.

Correct answer: B. (-4) and (-3)

(-\frac{22}{7}) is about (-3.14), so it lies between (-4) and (-3). In exams, note the order of negative decimals.

Correct answer: B. Seventh mark

When the interval is divided into 10 equal parts, each part has a value of \\(\frac{1}{10}\\). Since \\(0.7=\frac{7}{10}\\), the seventh mark from 0 represents 0.7. The eighth mark represents \\(0.8\\), so it is incorrect. In exams, convert the decimal into a fraction with denominator 10, 100, and so on to identify the mark.

Correct answer: C. 2.2

The exact midpoint of two numbers is their average: \(\frac{1.8+2.6}{2}=\frac{4.4}{2}=2.2\). Therefore, 2.2 is correct. The values 2.0 and 2.4 are not equally distant from both endpoints; for an exact midpoint, calculate the average of the two numbers.

Correct answer: A. 2.5

There are two numbers at a distance of 1.5 units from 4 on the number line: \\(4-1.5=2.5\\) and \\(4+1.5=5.5\\). Since \\(x<4\\), the number to the left of 4 must be chosen; hence, \\(x=2.5\\). Exam tip: Always check the direction or inequality condition in distance questions.

Correct answer: A. \\(1.25\\)

Moving to the right on a number line means adding the given distance to the starting number. Thus, \\(y=-2+3.25=1.25\\). Option D results from subtracting the distance, which would represent moving to the left. Exam tip: add for a rightward move and subtract for a leftward move.

Correct answer: C. (-\frac{5}{4})

(-\frac{5}{4}=-1.25), which is to the left of (-1). In exams, convert negative fractions into decimals to check.

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

(-\frac{3}{2}=-1.5), so it lies between (-2) and (-1). In exams, the decimal form of a negative fraction helps.

Correct answer: C. −5 and 5

From \(x^2-25=0\), we get \(x^2=25\). Therefore, \(x=5\) or \(x=-5\), so the solutions lie at −5 and 5 on the number line. Option B incorrectly includes 25; 25 is not a value of \(x\) satisfying the equation. Exam tip: When taking the square root of a positive number, consider both the positive and negative values.

Correct answer: A. (\frac{1}{2}) and (-4)

From (2x-1=0), (x=\frac{1}{2}), and from (x+4=0), (x=-4). In exams, solve each linear factor separately.

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

(\sqrt{2}) is about (1.414), so the order is (0,\frac{1}{2},\sqrt{2},2). In exams, increasing order is from left to right.

Correct answer: A. (x)

(-\sqrt{2}) is about (-1.414) and (-\frac{3}{2}=-1.5), so (x) is greater. In exams, for negative numbers, the point with smaller distance from (0) may lie to the right.

Correct answer: A. An irrational number is represented by a unique point on the number line and has a non-terminating, non-repeating decimal expansion.

Every real number corresponds to exactly one point on the number line. Irrational numbers such as \(\sqrt{2}\) have non-terminating, non-repeating decimals, so B and D are false. Exam tip: check whether an endless decimal repeats before classifying it.

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

Since \(5^2=25<26<36=6^2\), taking positive square roots gives \(5<\sqrt{26}<6\). Therefore, \(\sqrt{26}\) lies between 5 and 6. Exam tip: Compare the number with the two nearest consecutive perfect squares to locate its square root.

Correct answer: B. (\frac{7}{4})

Each part is (\frac{3}{12}=\frac{1}{4}), so the seventh point is (\frac{7}{4}). Divide the total length by equal parts.

Correct answer: C. 6.5

Since −√16 = −4, the distance is the absolute difference between the numbers: |2.5 − (−4)| = |6.5| = 6.5. Therefore, the correct answer is 6.5.

Correct answer: B. (-2.4)

(\sqrt{6}) is about (2.45), so (-\sqrt{6}) is about (-2.45). (-2.4) is greater, so it is to the right.

Correct answer: B. 0.6

The midpoint is the average of the two numbers: (−1.6 + 2.8) ÷ 2 = 1.2 ÷ 2 = 0.6. Therefore, option B is correct.

Correct answer: C. (7)

(\sqrt{45}) is about (6.7), so it is closer to (7). Nearby perfect squares (36) and (49) help in estimation.

Correct answer: A. -1.5

Moving left on the number line means subtracting the distance from the starting coordinate. Thus, the final coordinate is 1.25 - 2.75 = -1.5. Option C gives only the magnitude of the difference and ignores the negative direction. Exam tip: add for movement to the right and subtract for movement to the left.

Correct answer: A. (\frac{4}{3},\sqrt{2},1.5)

(\frac{4}{3}\approx1.33) and (\sqrt{2}\approx1.41), so the order is (\frac{4}{3}<\sqrt{2}<1.5). Estimate values for comparison.

Correct answer: C. (7) and (8)

The direct answer is option C: between 7 and 8. To locate a square root, compare the number under the root with consecutive perfect squares. We have \(7^2=49\) and \(8^2=64\). Since \(49<50<64\), taking positive square roots gives \(7<\sqrt{50}<8\). Option A, 5 and 6, is wrong because their squares are 25 and 36, both below 50. Option B, 6 and 7, is wrong because 50 is greater than \(7^2=49\), so the root is slightly greater than 7. Option C is correct. Option D, 8 and 9, is wrong because \(50<64=8^2\), so the root is below 8. Memory trick: bracket a square root using the two neighbouring perfect squares.