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Inequalities are mathematical expressions involving a comparison between two quantities. Unlike equations, which express equality, inequalities indicate a relationship where one side is either greater than, less than, greater than or equal to, or less than or equal to the other. For example, in the inequality \( y > 3x + 2 \), \( y \) is greater than \( 3x + 2 \). This inequality defines a region in the coordinate plane rather than a single line.
Within inequalities, terms use symbols such as:

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

When solving inequalities, the goal is to determine the range of values for the variable that satisfy the given conditions. It is essential to accurately solve and interpret inequalities as they are fundamental in describing real-world scenarios involving constraints and limitations.

The substitution method in coordinate geometry is a handy technique for verifying whether a point (or coordinate pair) satisfies an inequality or equation. To use this method, you replace the variable in the inequality or equation with the specific numbers given in the point.
This involves a couple of simple steps:

• First, identify the inequality or equation to check. In our example, it is \( y > 3x + 2 \).
• Then, substitute the given values of \( x \) and \( y \) into the inequality. For \((-2, 4)\), substitute \( x = -2 \) and \( y = 4 \) into \( y > 3x + 2 \) to get \( 4 > 3(-2) + 2 \).

By substituting, you determine if the values satisfy the inequality. It is a straightforward method that provides a clear path to verify solutions, especially useful when confirming points on graphs.

Graphing linear inequalities is a visual way to represent solutions on a coordinate plane. When you graph the inequality \( y > 3x + 2 \), you first graph the related equation \( y = 3x + 2 \) as a straight line. This line acts as the boundary of the inequality.

• Start by choosing points that satisfy the linear equation, then draw that straight line on the graph. Use a dashed line for inequalities that do not include the boundary \( (>, <) \), as the points on the line itself aren't solutions.
• Next, choose a test point not on the line, such as the origin \((0,0)\), and substitute it into the inequality to check if it satisfies the inequality. If it does, shade the region of the graph that includes this point. This shaded area represents all solutions to the inequality.

By graphing inequalities, you get a clear picture of where solutions lie relative to the boundary line. It visually separates the coordinate plane into two distinct regions: one that satisfies the inequality and the other that does not.