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The slope-intercept form is a way of expressing linear equations. It is given by the formula \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept. This form is particularly useful for graphing because it gives you two important pieces of information directly:

• The slope \(m\) tells you how steep the line is and in which direction it goes. For example, a positive slope means the line rises to the right, while a negative slope means it falls to the right.
• The y-intercept \(b\) gives you the starting point of the line on the y-axis. This is where the line crosses the y-axis.

In the exercise, we rearranged the inequality \(x - y < 5\) into \(y > -x + 5\), which aligns with the slope-intercept form. Here, \(m = -1\) and \(b = 5\). This indicates that the line has a downward slope with a y-intercept at the point (0,5). Understanding this form makes plotting the line on a graph straightforward.

The coordinate plane is a flat surface defined by horizontal and vertical lines called axes. The horizontal axis is known as the x-axis, and the vertical axis is the y-axis. Where these two axes intersect is called the origin, noted as point (0,0).When graphing, you place points on this plane using coordinates \((x, y)\), which tell you how far away from the origin to go horizontally and vertically. For the exercise, we plotted the line starting at the y-intercept (0,5) as given.

• Moving right on the x-axis, you follow the slope to determine where the line goes, using the slope \(-1\) to guide how much to go down.
• This creates a straight path, which is essential in depicting the general direction of the inequality.

The dashed line you drew reflects the inequality's open-ended solution, meaning values on the line are not included. This depiction on the coordinate plane visually represents solutions to the inequality.

Linear inequalities involve expressing a range of solutions where one variable is greater or lesser than another. For this exercise, we dealt with the inequality \(y > -x + 5\). Unlike linear equations, which result in a line where all points are solutions, inequalities allow for a region of solutions.

• To graph this inequality, you first plot what we call the boundary line \(y = -x + 5\), but since it's \(>\) not \(\geq\), we use a dashed line.
• Next, you shade the area where the inequality holds true, which in this case is above the line because it's \(y > -x + 5\).

This shading across the coordinate plane represents all the possible solutions which satisfy the inequality. It's crucial to test a point if unsure of which side to shade; often choosing something simple like the origin point (0,0), which if substituted in the inequality, tells if it lies within the solution region. Understanding this process is key to visualizing and solving various real-world problems that can be expressed through inequalities.