3. Solving Linear Inequalities in Two Variables

A linear inequality in two variables (like x + y ≤ 5) represents a region on the coordinate plane, not just a line.

A linear inequality in two variables is an inequality that involves two variables, typically x and y. Unlike a linear equation that represents a single line in a coordinate system, a linear inequality represents a range of values that satisfy the inequality, forming a shaded region in the graph. This region includes all the points (x, y) that make the inequality true.

Think of a situation where you need to create a mixture of two ingredients, x and y, and you have a limit on how much you can use. The inequality x + y ≤ 5 reflects the maximum combination of these ingredients you can use. In a graph, instead of just showing the exact combination, the shaded area shows all the possible combinations that fit within that limit.

🔹 Steps to Graph:

1. Rewrite the inequality in the form y = mx + c if needed.
2. Draw the boundary line:
   • Use a solid line for ≤ or ≥.
   • Use a dashed line for < or >.
3. Test a point (usually (0, 0)) to see which side of the line to shade.
4. Shade the region that makes the inequality true.

To graph a linear inequality, follow these steps:

1. Rewrite if Necessary: It helps to express the inequality in slope-intercept form (y = mx + c) to identify the slope (m) and y-intercept (c).
2. Draw the Boundary Line: Based on the inequality sign:
   • If the inequality is ≤ or ≥, draw a solid line, indicating points on the line are included in the solution.
   • If it is < or >, draw a dashed line, meaning points on this line are not part of the solution.
3. Test a Point: A common point to use is (0, 0). Substitute this point into the inequality to determine which side of the boundary line to shade.
4. Shade the Appropriate Region: Shade the region where the inequality holds true, showing all the possible solutions that satisfy the inequality.

Imagine you're designing a garden and can only use a limited amount of resources (like soil). The inequality represents how you can combine different shapes or areas of plants in your garden. The graph helps you visualize all the ways you can create spaces that fit within your resource limits, showing not just the boundary of what you can do, but all the options available to you.

🔹 Example: Graph x + y < 4

• Boundary line: x + y = 4 (draw dashed line)
• Test point: (0,0) → 0 + 0 = 0 < 4 → Shade the region including (0,0)

In this example, we are graphing the inequality x + y < 4. First, we rewrite it to identify the boundary line, which is x + y = 4. We draw a dashed line for this boundary because the inequality is 'less than' (<), meaning the line itself isn't included. Next, we test the point (0, 0): substituting gives us 0 + 0 = 0, which is less than 4. Since this point satisfies the inequality, we shade the area that includes (0, 0) and all points beneath the dashed line. This shaded area represents all solutions where x + y is less than 4.

Think of x + y as the total weight of fruits you can take in a basket. The line shows the limit of 4 kg of fruit. If (0, 0) is a point where you take no fruit, and it's allowed, you could fill your basket with any mix of apples and bananas as long as their combined weight stays below 4 kg. The shaded area shows all those combinations.

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