Question

question 1-26 in a new diet being researched, the number of fat units, x, and protein units, y, are to be increased or decreased from baseline values based on a relationship, for which some values are shown below. based on the values in the table, which inequality must hold? 10 -2 -4 2 0 x -1 6 8 5 4 y ○ y < 1/2 x + 4 ○ y < 1/2 x + 4 ○ y > 1/2 x + 4 ○ y ≥ 1/2 x + 4

Explanation:

Step1: Assume a linear - relationship

Let the linear relationship between \(x\) (fat units) and \(y\) (protein units) be \(y = mx + b\). Using two points \((x_1,y_1)=(0,4)\) and \((x_2,y_2)=(2,5)\) from the table. First, find the slope \(m\).
The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\), so \(m=\frac{5 - 4}{2-0}=\frac{1}{2}\).
Substitute \(x = 0\) and \(y = 4\) into \(y=mx + b\), we get \(4=\frac{1}{2}\times0 + b\), so \(b = 4\). The equation is \(y=\frac{1}{2}x + 4\).

Step2: Rearrange the equation to solve for \(x\) in terms of \(y\)

Starting with \(y=\frac{1}{2}x + 4\), subtract 4 from both sides: \(y - 4=\frac{1}{2}x\). Then multiply both sides by 2 to get \(x = 2y-8\), or \(y>\frac{1}{2}x + 4\) when considering the inequality relationship for the diet - model.

Answer:

\(y>\frac{1}{2}x + 4\)