Question

is the relationship between the values in the table to the right a direct variation, an inverse variation, or neither? if the relationship is a direct or inverse variation, then write an equation to model the variation. (table with x: 4, 8, 16, 22 and y: 10, 20, 30, 40) is the relationship between the values in the table a direct variation, an inverse variation, or neither? options: inverse, neither, direct (with neither selected) if the relationship is a direct or inverse variation, then write an equation to model the variation. choose the correct answer below. options: a. $y = \frac{22}{x}$, b. $y = 4x$, c. $y = 2.5x$, d. the data represent neither direct variation nor inverse variation. (with d as the correct option implied by the selected neither earlier)

Explanation:

Step1: Recall Direct and Inverse Variation

Direct variation: \( y = kx \) (constant ratio \( \frac{y}{x}=k \)). Inverse variation: \( y=\frac{k}{x} \) (constant product \( xy = k \)).

Step2: Check Direct Variation (Ratio \( \frac{y}{x} \))

• For \( x = 4, y = 10 \): \( \frac{10}{4}=2.5 \)
• For \( x = 8, y = 20 \): \( \frac{20}{8}=2.5 \)
• For \( x = 16, y = 30 \): \( \frac{30}{16}=1.875 \) (not equal to 2.5)
• For \( x = 22, y = 40 \): \( \frac{40}{22}\approx1.818 \) (not equal to 2.5)

Step3: Check Inverse Variation (Product \( xy \))

• For \( x = 4, y = 10 \): \( 4\times10 = 40 \)
• For \( x = 8, y = 20 \): \( 8\times20 = 160 \) (not equal to 40)
• For \( x = 16, y = 30 \): \( 16\times30 = 480 \) (not equal to 40)
• For \( x = 22, y = 40 \): \( 22\times40 = 880 \) (not equal to 40)

Since neither the ratio \( \frac{y}{x} \) (for direct) nor the product \( xy \) (for inverse) is constant, the relationship is neither.

Answer:

The relationship is neither. For the equation, the correct answer is D. The data represent neither direct variation nor inverse variation.