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The slope-intercept form is a way of expressing a linear equation so that it easily shows the slope and y-intercept of the line. It is written as \( y = mx + b \). Here, \( m \) represents the slope, which describes how steep the line is. It tells us how much the line rises or falls as it moves from left to right. The \( b \) term is the y-intercept, showing where the line crosses the y-axis.

To convert the inequality \( 3x - 2y \leq 6 \) into this form, we need to solve for \( y \). Start by subtracting \( 3x \) from both sides, resulting in \( -2y \leq -3x + 6 \). Next, divide every term by \(-2\) to isolate \( y \), which changes the inequality direction, yielding \( y \geq 1.5x - 3 \). Now, we have it in slope-intercept form, allowing us to identify a slope of \(1.5\) and a y-intercept of \(-3\). This makes graphing straightforward.

Plotting points involves placing them on a coordinate plane to represent a graph accurately. To plot the line from the inequality \( y \geq 1.5x - 3 \), we start by identifying the y-intercept at \(-3\) on the y-axis. This gives us our first point to mark on the graph.

Next, use the slope \(1.5\), which means for every unit moved to the right, move up 1.5 units. Starting from the y-intercept, move up 1.5 units and 1 unit to the right. Mark this second point on the graph. This slope method quickly helps in plotting linear equations. Draw a line through these points.

• For a positive slope, the line ascends; for a negative slope, it descends.
• The line is solid in this case, as the inequality is \( \geq \), indicating that points on the line satisfy the inequality.

Shading regions in a graph is crucial to show where solutions to an inequality lie. With the inequality \( y \geq 1.5x - 3 \), shading identifies all the points that satisfy it.

To decide which side of the line should be shaded, pick a test point not on the line, often the origin (0,0) is chosen if it's not on the line. Substitute (0,0) into the original inequality \( 3x - 2y \leq 6 \), and you get \( 0 \leq 6 \), which holds true. That tells us the side where (0,0) lies should be shaded. This method confirms the shading location ensures all points under the shading meet the inequality's conditions.

• Ensure that the region shaded makes the inequality true.
• When using test points, if the result is false, shade the opposite side.

Shading helps in visually understanding the solution set for the inequality.