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The coordinate plane is an essential tool for graphing inequalities as well as many other mathematical concepts. It consists of two perpendicular lines called axes: the horizontal axis, known as the x-axis, and the vertical axis, known as the y-axis. The point where these axes intersect is called the origin, denoted as \((0,0)\).

To effectively graph inequalities, it's important to understand the quadrants of the coordinate plane:

• Quadrant I: where both x and y coordinates are positive.
• Quadrant II: where x is negative, and y is positive.
• Quadrant III: where both x and y are negative.
• Quadrant IV: where x is positive, and y is negative.

Understanding these quadrants helps in identifying where the solution to an inequality might lie on the coordinate plane.

A linear equation in two variables, typically written in the form \(Ax + By = C\), represents a straight line on the coordinate plane. In the context of solving inequalities, you start by treating the inequality as though it is an equation, for instance \(x + y = 9\).

To graph this line, you can find two points that satisfy the equation:

• Set \(x = 0\) and solve for \(y\), which gives you the point \((0, 9)\).
• Set \(y = 0\) and solve for \(x\), which gives you the point \((9, 0)\).

With these two points, you can draw a straight line, which represents all solutions of the equation. This solid line will act as a boundary when graphing inequalities.

Remember that for inequalities, the line itself is not part of the solution when dealing with strict inequalities like \(<\).

Shading regions on the graph is crucial for representing solutions to inequalities. Once the boundary line is drawn, determine which side of the line represents the solution to the inequality.

To do this, select a test point not on the line, such as the origin \((0,0)\). Substitute this point into the inequality:

• If the value satisfies the inequality, shade the region where this point lies.
• If not, shade the opposite region.

For the inequality \(x + y < 9\), substituting \((0,0)\) gives \(0 + 0 < 9\), which is true. Therefore, the region that includes the origin is shaded.

This shaded area represents all the possible solutions to the inequality, where any point within will satisfy \(x + y < 9\). It's important to double-check by picking another point in the shaded area to see if it meets the inequality condition.