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The slope-intercept form is an essential way of expressing linear equations that is simple and straightforward. It allows us to quickly identify the slope and y-intercept of a line.
In this form, the equation of a line is written as \( y = mx + b \). Here, the variable \( m \) represents the slope, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
To rewrite an inequality such as \( x + 2y \leq 8 \) into slope-intercept form, start by solving for \( y \). Begin by subtracting \( x \) from both sides: \( 2y \leq 8 - x \).
Next, divide each term by 2 to isolate \( y \): \( y \leq 4 - 0.5x \). Now the inequality is in a form where the slope \( \text{(m)} = -0.5 \) and the y-intercept \( \text{(b)} = 4 \) are easily visible.

• Slope (m): Negative, indicating the line descends as it moves from left to right.
• Y-intercept (b): The line crosses the y-axis at (0, 4).

When graphing inequalities, the boundary line plays a critical role as it forms the edge of the solution set.
For the equation \( y = 4 - 0.5x \), this line divides the coordinate plane into two regions. It is important to decide whether to draw a solid or dashed line.The type of inequality determines the style of the line:

• Solid Line: If the inequality includes "or equal to" ( \( \leq \) or \( \geq \) ), the line is solid because points on this line satisfy the inequality.
• Dashed Line: If the inequality is strictly "less than" or "greater than" ( \( < \) or \( > \) ), the line is dashed, indicating points on the line do not satisfy the inequality.

In the inequality \( y \leq 4 - 0.5x \), the line is solid, as the inequality includes "or equal to." This means any point on this line is part of the solution.

Shading regions in graphing inequalities shows which side of the boundary line contains the solution to the inequality.
Identifying whether to shade above or below the line is key to visually representing the solution set.When dealing with a "less than or equal to" inequality, you will always shade the region below the line.
This represents all the points \((x, y)\) that satisfy the inequality \( y \leq 4 - 0.5x \).

• Shade Below: For \( y \leq \, \) you shade below the boundary line.
• Shade Above: For \( y \geq \, \) you shade above the boundary line.

In practice, pick a test point not on the line, perhaps \((0, 0)\), and check if it satisfies the inequality. If it does, you shade the side containing that point.