1. **Problem statement:** Find where the linear function $f(x)$ is less than zero ($f(x)<0$) and where it is greater than zero ($f(x)>0$). 2. **Formula and rules:** For a linear function $f(x) = ax + b$, the zero point (root) is found by solving $ax + b = 0$. 3. **Step 1: Find the zero point** Solve for $x$: $$ax + b = 0 \implies x = -\frac{b}{a}$$ 4. **Step 2: Determine sign intervals** - For $a > 0$, $f(x) < 0$ when $x < -\frac{b}{a}$ and $f(x) > 0$ when $x > -\frac{b}{a}$. - For $a < 0$, $f(x) < 0$ when $x > -\frac{b}{a}$ and $f(x) > 0$ when $x < -\frac{b}{a}$. 5. **Explanation:** The zero point divides the $x$-axis into two intervals. The sign of the slope $a$ determines on which side the function is positive or negative. 6. **Example:** Suppose $f(x) = 2x - 4$. - Find zero point: $$2x - 4 = 0 \implies 2x = 4 \implies x = \cancel{2}^{}$$ - Since $a=2 > 0$, - $f(x) < 0$ for $x < 2$ - $f(x) > 0$ for $x > 2$ **Final answer:** The function is negative for $x < -\frac{b}{a}$ and positive for $x > -\frac{b}{a}$, where $x = -\frac{b}{a}$ is the zero point of the function.