The absolute value of a number is its distance from zero on the number line, always positive or zero. The absolute value of $-\frac{7}{3}$ is the distance from $-\frac{7}{3}$ to 0, which is $\frac{7}{3}$ units. Therefore, $|-\frac{7}{3}| = \frac{7}{3}$.

We need to identify the greatest among -0.5, 0.1, -0.1, and 0.05. On the number line, positive numbers are greater than negative numbers, and among positive numbers, larger values are to the right. Comparing the positive values: 0.1 > 0.05, and both are greater than the negative values -0.5 and -0.1. Therefore, 0.1 is the greatest.

The absolute value of a number is its distance from zero on the number line, which is always positive or zero. The absolute value of $-\frac{9}{4}$ is the distance from $-\frac{9}{4}$ to 0, which is $\frac{9}{4}$ units. Therefore, $ \left| -\frac{9}{4} \right| = \frac{9}{4} $.

We need to identify the smallest among -2, -3/4, 0.5, and 1. Convert to decimals: -2 = -2, -3/4 = -0.75, 0.5 = 0.5, 1 = 1. On the number line, negative numbers are to the left of positive numbers, and among negative numbers, the one with greater absolute value is smaller. Since |-2| = 2 > |-0.75| = 0.75, -2 is smaller than -0.75.

We need to identify which expression represents a real number. √(-1) is undefined in real numbers (square root of negative), ln(-1) is undefined (logarithm of negative), $4^0$.5 = √4 = 2 is a positive real number, and 1/0 is undefined (division by zero). Only $4^0$.5 represents a real number.

This is an ordering of real numbers question testing number sense with irrational numbers. Choice B (3.14 < π < 22/7) is correct — converting to decimals: 3.14 = 3.1400..., π ≈ 3.14159..., 22/7 ≈ 3.14286. The correct order from least to greatest is: 3.14 < π < 22/7. Choice A (3.14 < 22/7 < π) places 3.14 correctly but swaps π and 22/7 — a very common error since 22/7 is often introduced as a shorthand for π, but it is actually slightly larger than π. Choice C (π < 3.14 < 22/7) incorrectly places π below 3.14, reversing their actual relationship — π ≈ 3.14159, which is greater than 3.14. Choice D (22/7 < π < 3.14) inverts the entire order, placing 22/7 as the smallest when it is actually the largest. Pro tip: Convert all three to decimals before comparing: 22 ÷ 7 ≈ 3.142857. This removes any ambiguity. Remember: 22/7 is a common approximation for π, but it overestimates π by about 0.001.

To approximate $\sqrt{3}$, we find perfect squares near 3. Since $1^2 = 1$ and $2^2 = 4$, $\sqrt{3}$ is between 1 and 2. More precisely, $1.7^2 = 2.89$ and $1.8^2 = 3.24$, so $\sqrt{3}$ is between 1.7 and 1.8. Calculating: $1.73^2 = 2.9929 \approx 3$, so $\sqrt{3} \approx 1.73$.

We need to identify which number is greatest among -5, -2, 0, and 3. On the number line from left to right: -5 < -2 < 0 < 3. The rightmost position represents the greatest value. Therefore, 3 is the greatest number. Choice A incorrectly thought -5 was greatest, confusing the magnitude of negative numbers with their actual value.

A rational number can be expressed as a fraction p/q where p and q are integers and q ≠ 0. √11 is irrational, 0.75 = 75/100 = 3/4 is rational (terminating decimal), π is irrational, and e is irrational. The decimal 0.75 terminates, making it rational.

We need to identify which number is greatest among -3.5, -2.75, -4, and -1.25. All numbers are negative, so the one closest to zero is greatest. On the number line from left to right: -4 < -3.5 < -2.75 < -1.25. The rightmost position represents the greatest value. Therefore, -1.25 is the greatest number. Choice A incorrectly selected -3.5, not understanding that among negative numbers, the one with smallest magnitude is greatest.