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The slope-intercept form is one of the most widely used forms for writing the equation of a straight line. It is expressed in the format \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept. The y-intercept is the point where the line crosses the y-axis. Understanding this form helps in easily identifying the slope and y-intercept, which are crucial for graphing lines.

The slope \(m\) describes how steep the line is. It is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. For the equation \(y = \frac{3}{4}x - 3\), the slope \(m\) is \(\frac{3}{4}\), which means for every 4 units you move to the right, the line rises by 3 units.

• Slope example: a slope of 2 means the line rises 2 units for every 1 unit it moves to the right.
• The y-intercept, \(-3\), signifies the line crosses the y-axis at \( (0, -3) \).

Solving inequalities involves finding all possible values that satisfy the inequality statement. In this context, inequalities differ from equations as they represent a range of solutions rather than a single line. For instance, the inequality \(y < \frac{3}{4}x - 3\) involves shading part of the graph to indicate all the solutions.

When graphing inequalities, it's important to determine whether to use a solid or dashed line. In this case, \(y < \frac{3}{4}x - 3\) is a "less than" inequality, so a dashed line is used to show that points on the line are not part of the solution.

• Dashed lines are used for \(<\) or \(>\).
• Solid lines are used for \(\leq\) or \(\geq\).

Once the line is drawn, the next step is to decide which side of the line represents the solution. Test a point not on the line, like \((0, 0)\), to see if it satisfies the inequality. If true, shade the region containing the tested point.

Graphing linear equations is the process of plotting a line based on its linear equation. When graphing, you'll typically start with the slope-intercept form. Plot the y-intercept first, which for the equation \(y = \frac{3}{4}x - 3\), is the point \((0, -3)\). Then use the slope to find another point.

Starting from the y-intercept, move up 3 units and right 4 units to plot another point that lies on the line. Connect these points using a straight line. Remember to determine if the line should be solid or dashed based on inequality rules. For the equation \(y < \frac{3}{4}x - 3\), a dashed line is used, to signify that points on the line do not belong to the solution set.

• Plot points based on y-intercept and slope.
• Connect with either a solid or dashed line, adhering to inequality indicators.
• Always verify by testing a point to ensure shading represents the correct set of solutions.