To find the area where a group of linear inequalities can work together, we need to draw these inequalities on a coordinate plane. This helps us see all the possible solutions that meet every inequality. Let’s break it down step-by-step!

Step 1: Draw Each Inequality

First, we need to graph each linear inequality on the same graph. A linear inequality looks like this:

$$y < mx + b$$

To graph it, turn the inequality (for example, $y < 2x + 1$) into an equation for the line that marks the boundary:

$$y = 2x + 1$$

1. Find Points: Pick at least two points on this line. For example:

   • When $x = 0$, $y = 1$ (point is $(0, 1)$).
   • When $x = -1$, $y = -1$ (point is $(-1, -1)$).

2. Draw the Line: Since our inequality is "less than" (<), use a dashed line. This shows that the points on the line aren’t included as solutions.

3. Shade the Area: Pick a test point that is not on the line (like $(0, 0)$). Check if this point satisfies the inequality. For instance, check $0 < 2(0) + 1$. This is true, so shade the area below the line. If it had been false, you’d shade above the line instead.

Step 2: Repeat for All Inequalities

Now, do the same for every inequality in your system:

1. For the inequality $y \geq -x + 3$, graph it like before. Since it’s “greater than or equal to,” draw a solid line to show the points on the line are included.
2. Shade the correct area using a test point again.

Step 3: Find the Feasibility Region

After you’ve graphed all the inequalities, look for the area where all the shaded regions overlap. This overlapping area is called the feasibility region. It represents all the possible solutions that meet every inequality in the group.

Example

Let’s check out a simple example:

1. $y < 2x + 1$
2. $y \geq -x + 3$

When you graph these, you end up with two lines that have different slopes:

• The line $y = 2x + 1$ goes up steeply and crosses the y-axis at (0, 1).
• The line $y = -x + 3$ goes down and crosses the y-axis at (0, 3).

You’ll see where their shaded areas overlap, making a shape that shows your feasibility region.

Final Thought

The feasibility region can be a closed shape (like a polygon) or it can stretch out endlessly (an unbounded area). If you find the region is empty, that means there are no solutions that work for all the inequalities. Be sure to check each inequality closely, and you’ll easily find the feasibility region. Happy graphing!