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Graphing inequalities involves shading a region on a graph where the inequality is true. Let's take a closer look at the inequality given: \( y \geq 3-x \). This inequality specifies all the points that lie on and above the line \( y = 3-x \). To graph this:

• First, graph the line \( y = 3-x \). This line has a slope of -1 and intersects the y-axis at 3.
• Since the inequality is \( \geq \), which includes equality, the line itself will be part of the solution set. This means you will draw a solid line.
• Next, to represent \( y \geq 3-x \), shade the region above this line, where the inequality holds. If you're unsure which side to shade, you can test a point that's not on the line, like (0,0), to see if it satisfies the inequality.

Shading the correct region shows visually where the solutions to the inequality exist.

The vertical line test is essential for determining whether a graph represents a function. This test ensures that for a graph to be a function, every x-value should correspond to exactly one y-value.Imagine drawing vertical lines across the graph. If any line you draw crosses the graph more than once, then the graph is not a function. This means there are multiple y-values for a single x-value, breaking the definition of a function.In our example of \( y \geq 3-x \), this region is shaded above a line. When you apply the vertical line test to this graph, you'll find that vertical lines will cross the shaded region at multiple points (except exactly on the line \( y = 3-x \)). This means the graph doesn't pass the test and therefore is not a function as defined by the typical standards.

Linear equations are fundamental in algebra and describe a straight line when plotted on a graph. The standard form of a linear equation is \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating the steepness and direction.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Using our equation \( y = 3-x \), we see it can be re-arranged to \( y = -x + 3 \), fitting perfectly in the standard form:

• The slope \( m = -1 \) suggests the line decreases as you move along the x-axis.
• The y-intercept \( b = 3 \) indicates the line crosses the y-axis at the point (0,3).

Linear equations like \( y = 3-x \) straightforwardly represent functions because they provide one y-value for every x. However, this function representation only holds for the line itself, not necessarily for shaded regions showing inequalities like \( y \geq 3-x \).