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Graphing inequalities on a coordinate plane helps visualize solutions for a set of equations. Each inequality divides the plane into two regions: one that satisfies the inequality and one that doesn't. To graph an inequality, follow these steps:

- Start by graphing the inequality as if it were an equation (use a solid line for \(≤\) or \(≥\) inequalities and a dashed line for \(<\) or \(>\) inequalities).
- Pick a test point (not on the boundary line) and substitute its coordinates into the inequality to see if it makes the inequality true.
- Shade the region where the inequality holds true.

For example, to graph \(5x - 3y < -2\), graph the line \(5x - 3y = -2\) using a dashed line, then shade below the line if the test point satisfies the inequality.

A solution to a system of inequalities is an ordered pair \(x, y\) that satisfies all inequalities in the system simultaneously. To determine if an ordered pair is a solution:

- Substitute the coordinates of the ordered pair into each inequality.
- Simplify the inequalities. If the pair satisfies each inequality, it is a solution.

For instance, consider the system:
\(5x - 3y < -2\)
and \(10x + 6y > 4\).

Testing the point \((1/5, 2/3)\):
- Substitute into \(5x - 3y < -2\), we get \(-1 < -2\) (False).
- Substitute into \(10x + 6y > 4\), we get \(6 > 4\) (True).
Because the point does not satisfy the first inequality, it is not a solution.

The substitution method is a way to determine if an ordered pair is a solution to a system of inequalities. It involves substituting the values of \(x\) and \(y\) into each inequality and simplifying to check their truth values. Steps to follow:

- Choose an ordered pair to test.
- Substitute the values of \(x\) and \(y\) from the pair into each inequality.
- Simplify each inequality to see if it holds true.

For the ordered pair \(\left(\frac{1}{5}, \frac{2}{3}\right)\), in the system of inequalities:
- For the first inequality \(5x - 3y < -2\), substitute \(x = \frac{1}{5}\) and \(y = \frac{2}{3}\), then simplify: \(1 - 2 < -2\). This is false, so the pair doesn't satisfy this inequality.

Ordered pairs \(x, y\) represent points on a coordinate plane. They are essential in checking solutions for systems of inequalities. Here's how they work:

- The first number in the pair is the \(x\)-coordinate, representing the horizontal position.
- The second number is the \(y\)-coordinate, representing the vertical position.

For example, to determine if \(\left(-\frac{3}{10}, \frac{7}{6}\right)\) is a solution, substitute:
- Into \(5x - 3y < -2\), giving \( -5 < -2\) (True).
- Into \(10x + 6y > 4\), giving \(4 > 4\) (False).
Since the pair does not satisfy both inequalities simultaneously, it is not a solution.