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The slope-intercept form of a linear equation is essential for graphing. It is written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) indicates the y-intercept, which is where the line crosses the y-axis. Understanding this form helps us visualize and sketch the line represented by a particular equation. When dealing with inequalities, we adjust the basic slope-intercept form slightly to reflect the inequality, such as \(y \leq mx + b\). This informs not only the position of the line but also which side contains the solutions to the inequality. Recognizing that the y-intercept gives a starting point on the graph, and the slope shows the rise over run, are foundational skills for graphing linear equations and inequalities effectively.

Linear inequalities are similar to linear equations, but instead of an equal sign, they use inequality symbols such as \(>\), \(<\), \(\geq\), or \(\leq\). This changes the meaning significantly. While an equation defines a straight line where all points on the line are solutions, an inequality describes a region of the plane that contains all solutions to the inequality. The inequality \(y \leq 4x + 10\) includes all points \((x, y)\) where \(y\) is less than or equal to \(4x + 10\). To graph it:

• We first plot the equivalent equation \(y = 4x + 10\) as a solid line (since it includes equality).
• The inequality dictates that we look below this line for all coordinates that satisfy \(y \leq 4x + 10\).

Recognizing these inequalities helps in visualizing large sets of potential solutions rather than focusing on distinct points.

Graphing linear inequalities requires a blend of plotting skills and understanding of inequalities. For the inequality \(y - 4x \leq 10\), after converting to slope-intercept form, \(y \leq 4x + 10\), we start by graphing the boundary line of the inequality. Here's how you can do it:

• Identify the y-intercept, which is 10 in this case, plotting the point \((0, 10)\).
• Use the slope (which is 4) to find another point. From \((0, 10)\), move up 4 units and 1 unit to the right to plot the next point \((1, 14)\).
• Draw a solid line through these points. The line is solid because \(y\) is less than or equal to \(4x + 10\).
• To determine the shaded area, select any test point not on the line (like \((0, 0)\)) and substitute into the inequality to test if it's true. Since \(0 \leq 4(0) + 10\) is true, shade the side of the line containing \((0, 0)\).

Remembering these steps simplifies graphing, letting you communicate the solution set visually.