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Linear inequalities are similar to linear equations but with inequality signs instead of equal signs. For example, the inequality can look like this: \(x + y \leq 4\).
Instead of having a single solution, they have solution sets that represent a range of values, often visible on a graph as a shaded area.

These inequalities can be written with different inequality signs: \(\leq\) for less than or equal to, \(\geq\) for greater than or equal to, \(\lt\) for less than, and \(\gt\) for greater than. This helps to determine which side of the boundary line to shade when graphing.
Understanding how to graph these inequalities correctly is crucial in determining solution sets.

Graphing linear inequalities involves several key steps. First, rewrite the inequality as an equation for graphing purposes. For instance, transform \(x + y \leq 4\) to \(x + y = 4\).
The next step is to find intercepts. Intercepts are the points where the graph crosses the axes. To find the x-intercept, set \(y = 0\) and solve for \(x\). To find the y-intercept, set \(x = 0\) and solve for \(y\). For our example, setting \(y = 0\) gives \(x = 4\) and setting \(x = 0\) gives \(y = 4\).
Draw a line through both intercepts, and since the inequality is \(\leq\) (less than or equal to), shade below the line.
This shaded region represents all the solutions to the inequality. Repeat this process for additional inequalities.

A solution set is the set of all possible values that satisfy a given inequality or system of inequalities. When graphing, it's represented by the shaded area.
For a system of inequalities, the solution set is found where the shaded regions of the individual inequalities overlap. This ensures all inequalities in the system are satisfied simultaneously.
In our example, after graphing \(x + y \leq 4\), \(x - y \leq 5\), and \(4x + y \leq -4\), the intersecting shaded region amongst all three represents the solution set.
To ensure accuracy, it's important to validate the solution by checking if specific points within the overlapping region satisfy all inequalities.

Intercepts are crucial in graphing as they provide exact points where the graph crosses the axes. There are two types of intercepts: x-intercepts and y-intercepts.
To find the x-intercept, set the equation's \(y\) value to 0 and solve for \(x\). For example, in the inequality \(x + y \leq 4\), setting \(y = 0\) gives \(x = 4\). Similarly, for the inequality \(4x + y \leq -4\), setting \(y = 0\) gives \(x = -1\).
To find the y-intercept, set the equation's \(x\) value to 0 and solve for \(y\). For example, in \(x + y \leq 4\), setting \(x = 0\) gives \(y = 4\). For the equation \(x - y \leq 5\), setting \(x = 0\) gives \(y = -5\).
Identifying these intercepts accurately helps in drawing precise boundary lines for each inequality, forming the basis for determining the solution sets on a graph.