91Ó°ÊÓ

The coordinate plane is a two-dimensional surface on which we can graph equations and inequalities. It consists of two perpendicular lines: the x-axis and the y-axis. These axes split the plane into four quadrants, making it easier to locate and plot points. Each point on the plane is identified by an ordered pair \((x, y)\), where \(x\) is the horizontal value and \(y\) is the vertical value.
To graph inequalities, start by drawing the coordinate plane and labeling the axes. Choose an appropriate scale for both axes to accurately reflect the values in your inequalities. This grid allows us to visually represent the solution set of an inequality by shading the appropriate region. When you graph a linear inequality, you're essentially creating a division of the plane into two regions: one that satisfies the inequality and one that does not.
The line in the graph represents the boundary between these regions and is drawn as either solid or dashed depending on the inequality symbol.

Slope-intercept form is a way of expressing linear equations, making it easier to graph them quickly. The general formula is \(y = mx + b\), where \(m\) represents the slope and \(b\) indicates the y-intercept.

• Slope (m): This is the rate of change, illustrating how much \(y\) changes for every unit increase in \(x\). A positive slope rises to the right, while a negative slope falls to the right.
• Y-intercept (b): This is where the line crosses the y-axis. At this point, \(x = 0\) and the value of \(y\) is \(b\).

When graphing a line using slope-intercept form, begin by plotting the y-intercept on the y-axis. Then, use the slope to find another point on the line by "rising and running" from the intercept: rise up or down by the numerator of the slope and run horizontally by the denominator. Connect these points to draw the line.

Linear equations form straight lines when graphed on a coordinate plane and have a constant rate of change. These equations are typically written in the form \(ax + by = c\) or transformed into slope-intercept form \(y = mx + b\) for ease of graphing.
Key features of linear equations include:

• They have no exponents higher than one.
• The graph of a linear equation is a straight line.
• Slope can be determined from any two points on the line using the formula \(m = \rac{y_2 - y_1}{x_2 - x_1}\).

Understanding how to manipulate linear equations into different forms is essential for solving and graphing inequalities, as it allows you to clearly identify the slope and intercept, making it easier to decide how to shade the graph.

Solving linear inequalities involves finding all the values of \(x\) and \(y\) that make the inequality true. When graphing, you start by treating the inequality as if it were an equation to find the line, then determine where to shade on the coordinate plane based on the inequality symbol.

• Solid Line: Used when the inequality symbol is \(\leq\) or \(\geq\), showing that points on the line satisfy the inequality.
• Dashed Line: Used when the symbol is \(>\) or \(<\), indicating that the line itself does not satisfy the inequality, only the area beyond it.
• Shaded Region: Represents all solutions to the inequality. For \(y >\) or \(y \geq\), shade above the line; for \(y <\) or \(y \leq\), shade below.

To verify which region to shade, you can select a test point (usually \(0,0\) if it's not on the line) and substitute it into the inequality. If it satisfies the inequality, shade that side; if not, shade the opposite side.