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The slope-intercept form is a standard way of expressing the equation of a line. It is written as \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating the steepness and direction. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

For example, in the inequality \( y \geq -x + 4 \), the equation of the line is \( y = -x + 4 \). Here,
the slope \( m \) is \(-1\) and the y-intercept \( b \) is 4, indicating that the line descends from left to right and crosses the y-axis at (0,4).
This form is particularly useful in graphing because it allows us to easily identify and plot the intercept point and determine the angle of the line.

When dealing with inequalities in the context of graphing, we are looking for a set of solutions that make the inequality true. This is usually represented as a region on a graph rather than as individual points or numbers.
For the inequality \( y \geq -x + 4 \), this forms a boundary line defined by \( y = -x + 4 \). However, because we are dealing with "greater than or equal to," the solution set includes not just the line itself but also all y-values that are greater.

• First, draw the line representing \( y = -x + 4 \). Use a solid line because the inequality includes equality (\( \geq \)).
• Then, determine which side of the line to shade. Here, you can test a point that does not lie on the line, like (0,0). Substitute these into the inequality: \( 0 + 0 \geq 4 \). This statement is false, so we shade the opposite side (where this is true).

Overall, shaded regions on the graph indicate all possible values that satisfy the inequality.

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane formed by two perpendicular axes: the horizontal x-axis and the vertical y-axis. It is used for graphing equations and inequalities, making it easy to visualize the solutions for problems involving two variables.

• The origin is the point where the x-axis and y-axis intersect, denoted as (0,0).
• Every point on a coordinate plane is defined by a pair of numbers (x, y), where x corresponds to the horizontal position and y to the vertical position.

When plotting an inequality such as \( y \geq -x + 4 \), you begin by positioning points on this graph based on the calculated line from the slope-intercept form. Using the y-intercept and slope, plot the line and identify which region represents the solution set by shading.
Practicing with the coordinate plane enhances understanding of spatial relationships and is an essential skill in algebra and many real-world applications.