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When working with systems of inequalities, you're dealing with two or more inequalities that share the same set of variables. Inequalities can be thought of as describing ranges of possible values instead of fixed values. For example, the inequality \(y \geq x\) signifies that for any value of \(x\), \(y\) must be equal to or greater than \(x\). Inequalities use symbols like \(>\), \(<\), \(\geq\), and \(\leq\) to show the relationship between two expressions.

• \(\geq\): greater than or equal to
• \(\leq\): less than or equal to
• \(>\): greater than
• \(<\): less than

When graphing inequalities, we often shade regions of the graph. These shaded regions show all the possible solutions that satisfy the inequality. For instance, shading above a line represents that all the points in that area make the inequality true.

Graphing linear equations is a key step when solving systems of inequalities. Each inequality can be represented as a line on a coordinate plane. The steps to graph these lines are:

• Find the slope and y-intercept of the line.
  For the equation \(y = mx + c\) where \(m\) is the slope and \(c\) is the y-intercept.
• Plot the y-intercept on the y-axis.
• Use the slope to find another point on the line.
  The slope tells you how to move from one point to another: rise over run.

For instance, in the inequality \(y \geq x\), the line \(y = x\) has a slope of 1 and intersects the y-axis at (0,0). To graph \(y \leq -x + 4\), draw the line \(y = -x + 4\) which has a slope of -1 and a y-intercept at (0, 4).
When you have the lines, determine whether to shade above or below. For example, \(y \geq x\) means you shade above the line, while \(y \leq -x + 4\) means you shade below the line.

After graphing the inequalities, the next step is to find the solution region. This is the area where the shaded regions of each inequality overlap. This overlapping area represents the set of all points that satisfy all inequalities in the system.
Steps to identify solution regions:

• Graph each inequality and shade the corresponding regions.
• Look for the overlap between the shaded regions.

For example, in the system
\[y \geq x\] and \(y \leq -x + 4\), the solution region is where the shaded area of \(y \geq x\) (above the line) intersects with the shaded area of \(y \leq -x + 4\) (below the line). Only the points in this overlapping area will satisfy both inequalities. The process of finding this region helps in visualizing and understanding how different linear constraints interact on a graph.