Line Intercept E6668E

1. **Problem Statement:** Given the x-intercept $(2,0)$ and y-intercept $(0,4)$, determine the equation of the line and analyze the intervals where the function is positive and negative. 2. **Formula and Rules:** The equation of a line with intercepts can be written as: $$\frac{x}{a} + \frac{y}{b} = 1$$ where $a$ is the x-intercept and $b$ is the y-intercept. 3. **Substitute the intercepts:** $$\frac{x}{2} + \frac{y}{4} = 1$$ 4. **Rewrite in slope-intercept form $y=mx+c$:** $$\frac{y}{4} = 1 - \frac{x}{2}$$ $$y = 4 - 2x$$ 5. **Analyze positivity and negativity:** - The function is positive when $y > 0$: $$4 - 2x > 0$$ $$4 > 2x$$ $$\cancel{2} \times 2 > \cancel{2} x$$ $$2 > x$$ So, $x < 2$. - The function is negative when $y < 0$: $$4 - 2x < 0$$ $$4 < 2x$$ $$2 < x$$ 6. **Check given intervals:** The user states positive for $x > 1$ and negative for $x < 1$, but from the equation, the function is positive for $x < 2$ and negative for $x > 2$. 7. **Final answer:** The equation of the line is: $$y = 4 - 2x$$ The function is positive for $x < 2$ and negative for $x > 2$.