In this image, we have two graphs showing linear inequalities that we need to express in slope-intercept form. Let's break down each graph:

Left Graph:

1. Identifying the slope and y-intercept:

   • The dashed line indicates that the inequality is either $<$ or $>$, meaning the boundary is not included.
   • The line crosses the y-axis at $y = 8$, so the y-intercept is $8$.
   • The line has a negative slope. To calculate the slope, we can use the points $(0, 8)$ and $(4, 0)$. $\text{slope} = \frac{0 - 8}{4 - 0} = \frac{-8}{4} = -2$
   • The inequality seems to shade below the line, so the inequality will be $y < -2x + 8$.

   Answer: $y < -2x + 8$

Right Graph:

1. Identifying the slope and y-intercept:

   • The solid line indicates that the inequality is either $\leq$ or $\geq$, meaning the boundary is included.
   • The line crosses the y-axis at $y = -2$, so the y-intercept is $-2$.
   • The slope is positive. Using the points $(0, -2)$ and $(2, 2)$, we calculate: $\text{slope} = \frac{2 - (-2)}{2 - 0} = \frac{4}{2} = 2$
   • The shading is above the line, so the inequality is $y \geq 2x - 2$.

   Answer: $y \geq 2x - 2$

Final Inequalities:

• Left graph: $y < -2x + 8$
• Right graph: $y \geq 2x - 2$

Would you like further details on how these inequalities were derived or have any other questions?

Here are some related questions:

1. How do you determine if a line should use $<$, $>$, $\leq$, or $\geq$?
2. What does the slope of a line represent in terms of the graph?
3. Why do we check the shading to determine the direction of the inequality?
4. How can you find the slope of a line using two points?
5. What changes if the inequality symbol is flipped (e.g., from $<$ to $\leq$)?

Tip: Remember that the type of line (solid or dashed) tells you whether the boundary is included in the inequality!