Graphing Inequalities 7B3135

1. **State the problem:** Graph the inequality $y \leq - \frac{1}{3} x + 2$. 2. **Formula and rules:** The boundary line is $y = - \frac{1}{3} x + 2$. Since the inequality is $\leq$, the boundary line is solid (included). Shade the region below the line because $y$ is less than or equal to the expression. 3. **Plot the boundary line:** - Find intercepts: - When $x=0$, $y=2$ (y-intercept). - When $y=0$, solve $0 = - \frac{1}{3} x + 2 \Rightarrow \frac{1}{3} x = 2 \Rightarrow x=6$. - Draw a solid line through points $(0,2)$ and $(6,0)$. 4. **Shade the solution region:** - Pick a test point not on the line, e.g., $(0,0)$. - Check inequality: $0 \leq - \frac{1}{3} \cdot 0 + 2 \Rightarrow 0 \leq 2$ (true). - Shade the region including $(0,0)$, which is below the line. --- 1. **State the problem:** Graph the inequality $y \leq \frac{1}{2} x - 5$. 2. **Formula and rules:** The boundary line is $y = \frac{1}{2} x - 5$. Since the inequality is $\leq$, the boundary line is solid. Shade below the line. 3. **Plot the boundary line:** - Intercepts: - $x=0 \Rightarrow y = -5$. - $y=0 \Rightarrow 0 = \frac{1}{2} x - 5 \Rightarrow \frac{1}{2} x = 5 \Rightarrow x=10$. - Draw solid line through $(0,-5)$ and $(10,0)$. 4. **Shade the solution region:** - Test point $(0,0)$: $0 \leq \frac{1}{2} \cdot 0 - 5 \Rightarrow 0 \leq -5$ (false). - Shade opposite side from $(0,0)$, which is below the line. --- 1. **State the problem:** Graph the inequality $y \geq \frac{1}{8} x + 2$. 2. **Formula and rules:** Boundary line $y = \frac{1}{8} x + 2$. Inequality $\geq$ means solid line and shading above. 3. **Plot the boundary line:** - Intercepts: - $x=0 \Rightarrow y=2$. - $y=0 \Rightarrow 0 = \frac{1}{8} x + 2 \Rightarrow \frac{1}{8} x = -2 \Rightarrow x = -16$. - Draw solid line through $(-16,0)$ and $(0,2)$. 4. **Shade the solution region:** - Test point $(0,0)$: $0 \geq \frac{1}{8} \cdot 0 + 2 \Rightarrow 0 \geq 2$ (false). - Shade opposite side from $(0,0)$, which is above the line. --- 1. **State the problem:** Graph the inequality $y > \frac{1}{4} x - 3$. 2. **Formula and rules:** Boundary line $y = \frac{1}{4} x - 3$. Inequality $>$ means dashed line (not included) and shading above. 3. **Plot the boundary line:** - Intercepts: - $x=0 \Rightarrow y = -3$. - $y=0 \Rightarrow 0 = \frac{1}{4} x - 3 \Rightarrow \frac{1}{4} x = 3 \Rightarrow x=12$. - Draw dashed line through $(0,-3)$ and $(12,0)$. 4. **Shade the solution region:** - Test point $(0,0)$: $0 > \frac{1}{4} \cdot 0 - 3 \Rightarrow 0 > -3$ (true). - Shade above the dashed line. **Final note:** The dot on the boundary lines for inequalities with $\leq$ or $\geq$ is solid to indicate inclusion.