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The boundary line is a crucial component when sketching regions defined by inequalities. To derive a boundary line, one must first take the inequality and express it as an equation by replacing the inequality symbol with an equal sign. For the inequality \(2x + y < 10\), the boundary line equation becomes \(2x + y = 10\).

• This line helps us divide the coordinate plane into distinct regions, with one region where the inequality holds true.
• It's important to note that the boundary line itself is not included as part of the solution if the inequality is strict ("<" or ">"), meaning the line is normally drawn dashed to indicate this.

Next, determine the intercepts to sketch the line accurately. Here, the x-intercept is found by setting \(y = 0\), giving us the point \((5,0)\). The y-intercept is found by setting \(x = 0\), resulting in the point \((0,10)\). Using these points, you can create a straight line that forms the boundary between the regions.

The coordinate plane is a two-dimensional surface on which we can plot equations and inequalities. It consists of two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0,0). This plane enables us to visualize solutions to equations or inequalities by marking points or shading regions.When plotting our inequality \(2x + y < 10\), the boundary line divides the plane into two half-planes. We must determine which side of this line satisfies the inequality. One effective strategy is using a test point, commonly \((0, 0)\), to verify which region should be shaded. In this instance, since \(0 < 10\), \((0, 0)\) satisfies the inequality, indicating that the region containing this point is the solution region. This visual depiction on the coordinate plane is invaluable for understanding relationships defined by inequalities.

Linear inequalities extend the concept of linear equations by introducing inequality symbols like "<", "\>" along with "\(leq\)" and "\(geq\)". Instead of fixing on a line of solutions, they describe a region in the coordinate plane that represents all possible solutions.

• For the inequality \(2x + y < 10\), it doesn't just describe a single path or line but a half-plane of solutions.
• These solutions are the collection of points that satisfy the inequality's conditions.

To find out which portion of the plane satisfies the inequality: - Draw the boundary line for \(2x + y = 10\) as a dashed line (since the inequality does not include the boundary line itself).- Choose a test point not on the boundary, such as \((0, 0)\). If \((0, 0)\) satisfies the inequality, then shade the side of the line where \((0, 0)\) resides.By understanding these steps, you can effectively visualize any linear inequality's solution on the coordinate plane as a shaded region, showing an infinite number of potential solutions within a designated area.