91Ó°ÊÓ

Understanding the slope-intercept form is crucial for graphing linear equations and inequalities. It allows us to write the equation of a line in the simple format of \( y = mx + b \), where \(m\) represents the slope and \(b\) is the y-intercept. Let's break this down:

The slope, \(m\), tells us how steep the line is and the direction it goes. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls. The y-intercept, \(b\), gives the exact point where the line crosses the y-axis.

In our exercise, we started with an inequality \(3x - 6y \leq 12\) and converted it to slope-intercept form to get \(y \geq -0.5x + 2\). This tells us our line will slope downward to the right because of the negative slope and it will cross the y-axis at \(2\).

Graphing an inequality follows similar steps to graphing an equation, but with a twist: we also need to demonstrate which side of the line contains solutions to the inequality. To do so, after plotting our line using slope-intercept form, we choose a shading technique that conveys where the inequality holds true.

With our inequality, \(y \geq -0.5x + 2\), once we've plotted the line, we determine which side to shade. The symbol \(\geq\) indicates that the solutions include points on the line and the area above it. If our inequality had the symbol \(<\) or \(\leq\), we would shade the area below the line. Finally, use a solid line to represent \(\geq\) or \(\leq\), and a dashed line for \(>\) or \(<\), since the solid line includes points on the line as part of the solution set, while the dashed line does not.

Plotting points is a fundamental skill in graphing. Every point on a coordinate plane is defined by an ordered pair (x, y), where 'x' indicates horizontal position, and 'y' vertical. To plot a point, start at the origin (0,0), then move along the x (horizontal) axis according to the first number, and follow by moving parallel to the y (vertical) axis as per the second number.

In our problem, the y-intercept provided a starting point: (0,2). From here, the slope influenced how we plotted additional points. For our negative slope of -0.5, starting from the y-intercept, we moved 1 unit to the right (positive x direction) and 0.5 units down (negative y direction) to plot our next point. This systematic approach, using slope and y-intercept from slope-intercept form, simplifies creating a visual representation of linear inequalities on a graph.