3.1. Steps to Graph

1. Rewrite the inequality in the form y = mx + c if needed.

Before graphing, it's important to rearrange the inequality so that y is isolated on one side of the equation. This means putting it into slope-intercept form (y = mx + c), where 'm' represents the slope, and 'c' is the y-intercept. This form makes it easy to identify how the line behaves on a graph.

Think of it like organizing a recipe. Just as you might write 'Add flour' instead of 'Flour added,' here we write the inequality in a clear, structured way to see how it will look on a graph.

2. Draw the boundary line:

• Use a solid line for ≤ or ≥.
• Use a dashed line for < or >.

The boundary line represents where the inequality equals the value. If the inequality uses '≤' or '≥', we draw a solid line because points on this line are included in the solution. Conversely, for '<' or '>', we use a dashed line, indicating that points on the line are not part of the solution.

Imagine setting a speed limit. If the sign says 'You may drive at speeds less than 60mph,' the 60mph point is not included, just like a dashed line. But if it says 'You may drive at speeds up to 60mph,' the 60mph point is included, like a solid line.

3. Test a point (usually (0, 0)) to see which side of the line to shade.

After drawing the boundary line, pick a test point to determine which side of the line to shade. A common choice is the origin point (0, 0), unless this point lies on the line. Substitute this point into the original inequality. If the statement is true, shade the side that includes the test point; if false, shade the opposite side.

Think of this step like checking if a light switch is on or off. You flip the switch (test a point) to see if it is lighting up the room (the area to shade) or if it’s not (the area to leave unshaded).

4. Shade the region that makes the inequality true.

After determining which side of the line to shade, use a pencil or color to shade that area of the graph. This shaded region represents all the possible solutions to the inequality, where any point in this area satisfies the original inequality.

Imagine coloring a map to show where it's safe to walk. The shaded area is like the safe zone, where you can be assured that any point you pick in this area meets your criteria for safety.

🔹 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)

For the inequality x + y < 4, we first find the boundary line by rearranging to y = -x + 4. We then sketch a dashed line for this boundary because it uses '<'. Next, we test the point (0, 0) by substituting these values into the inequality, which yields true (0 < 4). Therefore, we shade the area that includes (0, 0), which is the region where all points satisfy the initial inequality.

This is akin to a graphic showing where temperatures are below a certain limit. The dashed line shows the threshold temperature, and the shaded area represents all the temperatures that are permissible, with (0,0) being a clear example of a permissible temperature.

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