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Linear equations are the backbone of algebra. They describe a straight line when graphed on a coordinate plane. A standard form for a linear equation is \( y = mx + b \) where \( m \) represents the slope of the line and \( b \) is the y-intercept. This means \( b \) is where the line crosses the y-axis. You can think of the slope, \( m \), as the 'rise over run', or how much the line rises vertically for every step it moves horizontally. When graphing, start at the y-intercept and use the slope to find other points on the line. For example, if \( m = -1 \) (a negative slope), the line will descend as you move from left to right. This is exactly what happens in the function \( f(x) = 4 - x \). Here, the line starts at 4 on the y-axis and heads downward with a slope of \(-1\).
Linear equations are simple yet powerful and understanding them is crucial to mastering topics involving graphing and functions.

Inequalities are like equations, but instead of an equal sign, they use symbols like \( <, >, \leq, \geq \) to compare two expressions. When you see an inequality like \( f(x) \geq 0 \), it means you're looking for values of \( x \) where the expression is possibly zero or positive. Solving inequalities often means finding a range or interval of values. In our exercise, after graphing the function and setting \( f(x) = 0 \), we found that \( x = 4 \) is a key point.
From the graph, if the line is above the x-axis, \( f(x) \geq 0 \). For the function \( f(x) = 4 - x \), this happens when \( x \) is less than or equal to 4, so the solution is the interval \([-∞, 4]\). Inequalities are important in representing real-world scenarios, like finding minimum requirements or thresholds.

Graphing a function involves sketching the function on a coordinate plane to visualize its behavior. For linear functions like \( f(x) = 4 - x \), the graph is a straight line. Start by finding the y-intercept, which is where \( x = 0 \). For \( f(x) = 4 - x \), plug \( x = 0 \) into the equation to get \( f(0) = 4 \).
Next, plot this point \((0,4)\) on the graph. Use the slope \(-1\) to determine the direction and steepness of the line. Move one unit right and one unit down from the y-intercept to plot a second point. Connect these points, ensuring the line extends beyond the given points for a complete representation.
Finally, analyzing the graph helps determine where \( f(x) \geq 0 \). The line intersects the x-axis at the solution to \( f(x)=0 \), which is \( x=4 \). The line is above the x-axis for \( x < 4 \), confirming the interval \([-∞, 4]\). Understanding and graphing functions is essential for visualizing mathematical relationships.