Identify/Shade the feasible region.4x + 2y ≤ 32

x + 3y < 18

To identify and shade the feasible region for the system of inequalities:

1. **Inequality 1**: \( 4x + 2y \leq 32 \)
   
   - First, rewrite this inequality in slope-intercept form (\( y = mx + b \)) by solving for \( y \):
   
   \[
   4x + 2y \leq 32
   \]
   \[
   2y \leq -4x + 32
   \]
   \[
   y \leq -2x + 16
   \]
   
   This is the equation of a line with slope \( -2 \) and y-intercept \( 16 \), and the region below or on this line is part of the feasible region.

2. **Inequality 2**: \( x + 3y < 18 \)
   
   - Similarly, solve for \( y \):
   
   \[
   x + 3y < 18
   \]
   \[
   3y < -x + 18
   \]
   \[
   y < -\frac{1}{3}x + 6
   \]
   
   This is the equation of a line with slope \( -\frac{1}{3} \) and y-intercept \( 6 \), and the region below this line is part of the feasible region.

### Steps to graph the feasible region:

1. Graph the line \( y = -2x + 16 \) (solid line) and shade below the line because of the \( \leq \) inequality.
2. Graph the line \( y = -\frac{1}{3}x + 6 \) (dashed line) and shade below the line because of the \( < \) inequality.
3. The feasible region is the area where both shaded regions overlap.

Would you like an illustration of this graph and feasible region? Let me know if you need more details or have questions!

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Here are 5 questions to expand on the concept:

1. How do you find the intersection points of the lines in a system of inequalities?
2. What does the slope of a line tell you about the direction of shading for an inequality?
3. How do you determine whether to use a solid or dashed line when graphing inequalities?
4. What would happen if one of the inequalities had a strict equality (\( = \)) instead of an inequality sign?
5. How does changing the inequality from \( \leq \) to \( \geq \) affect the shaded region?

**Tip:** Always plot the boundary lines first, then decide which side to shade by testing a point!

Identify/Shade the feasible region. 4x + 2y ≤ 32, x + 3y < 18

Learn how to identify and graph the feasible region for the system of inequalities 4x + 2y ≤ 32 and x + 3y < 18. This detailed guide covers key algebraic concepts, including rewriting inequalities in slope-intercept form, graphing the boundary lines, and shading the overlapping region. Ideal for students in Grades 9-11.

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