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The slope-intercept form is an efficient way to express linear equations. It is typically written as \(y = mx + c\), where:

• \(m\) is the slope of the line, representing the steepness. It shows how much \(y\) changes with \(x\).
• \(c\) is the y-intercept, indicating where the line crosses the y-axis.

This format helps us quickly identify how to plot the line on a graph. For example, in the inequality \(y < \frac{2}{3}x - 1\), the slope \(m\) is \(\frac{2}{3}\). This tells us that for every 3 units we move to the right on the x-axis, we move 2 units up on the y-axis. The y-intercept \(c\) is \(-1\), so our graph line will cross the y-axis at \(y = -1\). This straightforward approach makes graphing linear equations much easier.

Inequalities are mathematical expressions used to indicate when one value is not equal but instead greater or less than another. To graph an inequality like \(y < \frac{2}{3}x - 1\), we start by understanding the basic equation \(y = \frac{2}{3}x - 1\). Here,

• 'less than' implies the shaded area will be below the line if \(y\) values are not included on the line itself.
• The boundary line is often drawn as a dashed line to signify that points on the line are not included in the solution.

In inequalities, it's crucial to determine whether the line should be solid or dashed, as this directly impacts which regions are part of the solution. "Less than" or "greater than" symbols \((<, >)\) lead to a dashed line, while "less than or equal to" or "greater than or equal to" \((\leq, \geq)\) results in a solid line.

Graphing linear equations involves plotting points on a graph to display the relationship between two variables. The foremost step is transforming the equation into the slope-intercept form \(y = mx + c\). For the inequality \(y < \frac{2}{3}x - 1\), begin with drawing the line for \(y = \frac{2}{3}x - 1\).

• Start by marking the y-intercept, \(c\) at \(-1\) on the y-axis.
• Next, use the slope \(\frac{2}{3}\) to identify another point. From the y-intercept, move right 3 units and up 2 units to plot another point.

Draw a line through these points, ensuring it's dashed because we're dealing with a "less than" inequality. Your graph should include shading below the line, indicating all \(y\) values that satisfy \(y < \frac{2}{3}x - 1\). Thus, graphing linear inequalities effectively translates mathematical expressions into visual representations. This allows us to see all possible solutions.