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Graphing inequalities involves representing inequalities on a graph and identifying the regions that satisfy them. To start, you need to plot the boundary lines of the inequalities. This is done by converting each inequality into an equation. For instance, take the inequality \(0.2x + 0.3y \geq 7.2\) and convert it to the equation \(0.2x + 0.3y = 7.2\). Do the same for \(2.5x - 6.7y \leq 0\), creating the equation \(2.5x - 6.7y = 0\).
To plot these equations, simply draw straight lines on a Cartesian plane. These lines represent the bounds or limits of the areas satisfying each inequality. Once you have your boundary lines on the graph, use a test point, like \((0, 0)\), to determine whether the region above or below the line satisfies each inequality. Shade the regions that meet the inequality conditions to visually represent the solution set.
Graphing inequalities helps in visualizing solutions and understanding where systems of inequalities overlap.

Systems of inequalities are multiple inequalities that must be satisfied at the same time. Just like systems of equations, these inequalities are solved against one another. By graphing each inequality, you create multiple boundary lines, each dividing the plane into regions. The solution to the system is the overlapping region where all these conditions are met.
For example, a system including \(0.2x + 0.3y \geq 7.2\) and \(2.5x - 6.7y \leq 0\) needs both the regions representing \(x\) and \(y\) values that satisfy these constraints.
To identify the common solution area, carefully analyze the graph. The graph will show a overlapped region where the shading for each inequality intersects. This region represents every point \((x, y)\) that satisfies both inequalities simultaneously.
Knowing which side of the boundary each inequality's solution lies on helps in finding the overlapping region quickly and accurately.

Intersection points are where the graphs of the boundary lines of the inequalities cross, indicating potential solutions for the systems of inequalities. In the context of graphing inequalities, these points are significant because they define the corners of the region where all conditions are met.
To find intersection points, solve the system of equations given by the boundary lines. For the equations \(0.2x + 0.3y = 7.2\) and \(2.5x - 6.7y = 0\), apply either substitution or elimination methods to find the exact coordinate point \((x, y)\).
Once calculated, verify that this point lies within the shaded region. If it is, then the intersection point is valid and part of the solution. Rounding these coordinates to two decimal places ensures a precise answer, especially when using technology.

Boundary lines separate different regions on the graph and are crucial in graphing inequalities. They form the visual limits of the areas where the inequalities hold true. Each boundary line corresponds to an equality derived from the original inequality by replacing the inequality sign with an equal sign.
For the inequality \(0.2x + 0.3y \geq 7.2\), the boundary line is \(0.2x + 0.3y = 7.2\), and for \(2.5x - 6.7y \leq 0\), it is \(2.5x - 6.7y = 0\). These lines help in determining which side the inequality holds. Use test points to decide which side of each boundary line is the feasible region.
When graphing, respect whether the inequality is strict (\(<\) or \(>\)) resulting in a dashed line, or inclusive (\(\leq\) or \(\geq\)), depicted by a solid line. This distinction helps understand if the points on the line are included in the solution. Ultimately, boundary lines help create a clearer visualization of where solutions lie on the plane.