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When dealing with inequalities, finding solutions means identifying points on a graph that satisfy the inequality condition. To check if a point, such as \((0,0)\), is a solution, substitute the coordinates into the inequality. If the inequality holds true, that point is a solution. In our example, when we substitute \(x=0\) and \(y=0\) into the inequality \(3x - y < 3\), we get \(0 < 3\). This statement is true, confirming that \((0,0)\) is indeed a solution to the inequality.

For any inequality solution, the aim is to determine a range of values or points that satisfy the inequality. In graphical terms, this involves shading a region of the graph where all solutions lie. To fully graph the inequality, it's important first to sketch the boundary line, then determine which side of this line contains the solutions.

The Cartesian plane is a two-dimensional surface defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is represented by a coordinate pair \((x,y)\). The Cartesian plane allows us to graphically represent equations and inequalities. For our inequality \(3x - y < 3\), plotting it helps visualize all the possible solutions.

When graphing an inequality, start by drawing the line that represents the boundary. For \(3x - y = 3\), this line separates the solutions from non-solutions. Remember:

• Use a solid line for inequalities that include equality (\(\leq\) or \(\geq\)).
• Use a dashed line for strict inequalities (\(<\) or \(>\)).

The choice of line type indicates whether points on the line are included as solutions.

Coordinate graphing involves plotting points, lines, and regions on the Cartesian plane. In this case, we convert the inequality \(3x - y < 3\) into the equation \(3x - y = 3\) to graph the line. First, rearrange to \(y = 3x - 3\), which is in slope-intercept form \(y = mx + b\).

Here's how to graph it:

• Identify the y-intercept \(b\) which is \(-3\), and plot this on the y-axis.
• Determine the slope \(m\), which is 3, meaning for every unit increase in \(x\), \(y\) increases by 3.
• Draw the line using these elements.

Since the inequality is \(3x - y < 3\), the line should be dashed, indicating points on the line aren’t part of the solutions. Choose a test point (like \((0,0)\)) to see which side of the line satisfies the inequality, and shade that region. This shaded area represents all the possible solutions where the inequality holds true.