1. **State the problem:** We are given a system of inequalities and points to check, and then asked to graph inequalities and describe graph shapes. 2. **Check points for inequalities:** - For $x + y < 7$ and point $(2,3)$: Substitute $x=2$, $y=3$: $$2 + 3 = 5 < 7$$ Point satisfies the inequality. - For $x + 3y \geq -2$ and point $(-9,1)$: Substitute $x=-9$, $y=1$: $$-9 + 3(1) = -9 + 3 = -6 \geq -2?$$ No, $-6$ is not greater than or equal to $-2$, so point does not satisfy. - For $-6x + 4y \leq 6$ and point $(-3,-3)$: Substitute $x=-3$, $y=-3$: $$-6(-3) + 4(-3) = 18 - 12 = 6 \leq 6$$ Point satisfies the inequality. 3. **Graphing inequalities:** - For $y \leq 5$: This is a horizontal line at $y=5$ with shading below or on the line. - For $x < 2$: This is a vertical line at $x=2$ with shading to the left (less than 2). - For $y \geq -x - 1$: Line with slope $-1$ and y-intercept $-1$, shading above or on the line. - For $5x - 2y \leq 6$: Rearrange to slope-intercept form: $$5x - 2y \leq 6 \Rightarrow -2y \leq 6 - 5x \Rightarrow y \geq \frac{5x - 6}{2}$$ So line with slope $\frac{5}{2}$ and y-intercept $-3$, shading above or on the line. - For $-x + 4y > -12$: Rearrange: $$4y > x - 12 \Rightarrow y > \frac{x}{4} - 3$$ Line with slope $\frac{1}{4}$ and y-intercept $-3$, shading above the line (not including line). - For $\frac{4}{3}x + y < 0$: Rearrange: $$y < -\frac{4}{3}x$$ Line with slope $-\frac{4}{3}$ and y-intercept $0$, shading below the line. 4. **Graph shapes descriptions:** - Left graph for $y < -x + 1$: Line with slope $-1$, y-intercept $1$, shading below the line. - Right graph for $y \leq 3x - 2$: Line with slope $3$, y-intercept $-2$, shading below or on the line. 5. **Summary:** - Points satisfy or do not satisfy inequalities as checked. - Inequalities are graphed as lines with shading indicating solution regions. **Final answers:** - $(2,3)$ satisfies $x + y < 7$. - $(-9,1)$ does not satisfy $x + 3y \geq -2$. - $(-3,-3)$ satisfies $-6x + 4y \leq 6$. - Graphs are lines with slopes and intercepts as described, with shading indicating solution regions.