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When we talk about linear inequalities, we're working with expressions that involve a linear relationship between variables—usually 'x' and 'y'—and these relationships are not equalities but inequalities. This means instead of saying 'y' is exactly equal to a line (as in 'y = mx + b'), we're saying 'y' is less than, greater than, less than or equal to, or greater than or equal to a line. For example, a simple linear inequality might look like 'y > 2x + 1'.

In the context of a coordinate plane, each linear inequality, like 'y ≤ -3x + 5', corresponds to one side of the boundary line drawn by the equation 'y = -3x + 5'. The inequality sign tells us whether to shade above or below this line. It's crucial in a system of inequalities to understand how to analyze these signs and know which side of the boundary line to shade because this represents the set of all possible solutions that satisfy the inequality.

Graphing linear inequalities is a common method to visualize the solution set. The exercise provided, however, focuses on analyzing such inequalities algebraically without graphing. When multiple inequalities are combined, as in a system, we are looking for common areas that satisfy all the inequalities simultaneously.

In systems of equations, we often find a single solution (a point), many solutions (a line or area), or no solution at all. This is also true for systems of inequalities. A system of linear inequalities will have no solution if the areas represented by the inequalities have no overlap. In essence, if there's no shared region that satisfies all inequalities, no set of (x, y) coordinates will solve the system.

Conversely, we say a system has infinitely many solutions when there's a limitless set of (x, y) pairs that satisfy all the inequalities at once. This typically occurs in a system when the inequalities are the same or are essentially the same line with an 'equal to' component. In the given system, both inequalities describe the same line and encompass all points on either side due to the 'less than or equal to' and 'greater than or equal to' signs. This remarkably means every point in the plane is a solution to the system, indicating infinitely many solutions.

Determining if there's no solution or infinitely many solutions without graphing requires a careful analysis of the equations' properties and an understanding of how inequalities work together within a system.

Boundary line analysis is a critical step in understanding systems of inequalities. The boundary line of an inequality divides the coordinate plane into two halves. One half contains the solutions to the inequality; the other does not. The key to boundary line analysis is identifying where this dividing line is and which side of it is relevant to the inequality.

In our example, the boundary line analysis would lead us to the equation '6x - y = 24', which is a straight line. When the inequality signs ≤ and ≥ are applied, as in our system '6x - y ≤ 24' and '6x - y ≥ 24', we look at both sides of this line. Because one inequality includes all points on one side and the other includes all points on the opposite side (and the line itself), the entire plane is involved.

Notably, the exercise provided does not require graphing, but understanding boundary lines is still critical. Recognizing that both inequalities share the same boundary line helps identify the solution set without the need to visualize it. When both inequalities encompass all spaces on either side of the boundary line, as in this scenario, we have a clear case of a system with infinitely many solutions.