https://file.aiquickdraw.com/mathbot/file/17265983147164t850a5k.jpg

open curly brackets table attributes columnalign left end attributes row cell y space greater or equal than space minus 4 x space plus space 1 end cell row cell y space less or equal than space minus 4 x space plus space 4 end cell end table close

A. open curly brackets table attributes columnalign left end attributes row cell y space less or equal than space minus 4 space plus space 1 end cell row cell y space greater or equal than space minus 4 x space plus space 4 end cell end table close

B. open curly brackets table attributes columnalign left end attributes row cell y space less than space minus 4 x space plus space 1 end cell row cell y space greater than space minus 4 x space plus space 4 end cell end table close

D. open curly brackets table attributes columnalign left end attributes row cell y space greater than space minus 4 x space plus space 1 end cell row cell y space less than space minus 4 x space plus space 4 end cell end table close

The systems you have seen thus far all have inequalities in which the boundary lines intersect. Take a look at the graph, which shows parallel boundary lines.

As with intersecting boundary lines, if the inequality symbol is less than  or less or equal than , shade below the boundary line. If the inequality symbol is greater than or greater or equal than , shade above the boundary line. The solutions are all of the ordered pairs in the overlapping shaded area.

The image shows a graph with two parallel boundary lines, both with a slope of -4. The inequalities related to these lines appear to represent a system where we are solving for overlapping shaded regions.

Let’s interpret the inequalities given:

1. **Inequality 1**: \( y \geq -4x + 1 \)
   - This is a line with a slope of -4 and a y-intercept at (0, 1). The inequality tells us to shade above this line.

2. **Inequality 2**: \( y \leq -4x + 4 \)
   - This line also has a slope of -4 but a different y-intercept at (0, 4). The inequality tells us to shade below this line.

The solution is the region where the shaded areas from both inequalities overlap. This region is between the two parallel lines.

Now, let's compare these findings with the choices presented:

A. The inequalities in A are:
   - \( y \leq -4x + 1 \)
   - \( y \geq -4x + 4 \)

B. The inequalities in B are:
   - \( y < -4x + 1 \)
   - \( y > -4x + 4 \)

C. The inequalities in D are:
   - \( y > -4x + 1 \)
   - \( y < -4x + 4 \)

The correct system of inequalities in the graph should be:

\[
\boxed{y \geq -4x + 1 \quad \text{and} \quad y \leq -4x + 4}
\]

This corresponds to option **A**, as it correctly represents the region between the two parallel lines, shading above the lower line and below the upper line.

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Would you like further clarification or details about this? Here are five questions that may help deepen your understanding:

1. What are the rules for determining which region to shade in a graph of inequalities?
2. How do you find the slope and intercept from an inequality equation?
3. Can you explain the difference between strict inequalities (>, <) and non-strict inequalities (≥, ≤)?
4. What happens when boundary lines intersect in a system of inequalities?
5. How would the graph look if we swapped the inequality signs in the system?

**Tip:** When graphing inequalities, remember that solid lines are used for "greater than or equal to" (≥) or "less than or equal to" (≤), and dashed lines are used for strict inequalities (> or <).

Determine the solution to the system of inequalities y ≥ -4x + 1 and y ≤ -4x + 4, and identify the overlapping shaded region between these two parallel lines.

This problem involves solving a system of linear inequalities with parallel lines. Learn how to graph y ≥ -4x + 1 and y ≤ -4x + 4 and find the solution in the overlapping region. Suitable for students in Grades 8-10.

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