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When we talk about linear inequalities, we're extending the concept of a linear equation to an expression that includes inequality symbols like <, >, \( \leq \) or \( \geq \). Essentially, a linear inequality looks very much like a linear equation, except it defines a range of values instead of one exact solution. The solution to a linear inequality is a set of values for the variable that makes the inequality true.

For example, the inequality \( y \leq \frac{1}{4}x \) signifies a set of points \( (x, y) \) on a graph that are located on the line \( y = \frac{1}{4}x \) or below it. In these types of problems, we are often tasked with graphing the solution set to visualize the range of possible solutions. It's critical to remember that the line itself can be part of the solution if the inequality symbol includes an equal sign, as it does in this case.

The slope-intercept form of a line is a straightforward way to read and graph linear equations. This form is expressed as \( y = mx + b \) where \( m \) is the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis. The slope represents the steepness of the line and is calculated as the rise over run. In our example \( y = \frac{1}{4}x \) the slope is \( \frac{1}{4} \) which means a rise of 1 unit in y for every 4 units of increase in x.

The y-intercept, in this case, is 0, identified in the equation as just \( y = \frac{1}{4}x \) without an added constant. This means our line crosses the y-axis at (0, 0). To graph a line in slope-intercept form, first plot the y-intercept, and then use the slope to find a second point. From there, you can draw the line through these two points.

Once you have graphed the line for a linear inequality, the next step is to shade the correct region of the graph. This is known as inequality shading and it visualizes all the possible solutions to the inequality. The direction of shading is determined by the inequality symbol:

• If the symbol is \( \leq \) or <, we shade below the line.
• If the symbol is \( \geq \) or >, we shade above the line.

For the inequality at hand, \( y \leq \frac{1}{4}x \), we shade below the line, which includes every point that has a y-value less than or equal to \( \frac{1}{4} \) times the corresponding x-value. This indicates all solutions to the inequality. It's important to shade lightly so that the line remains visible, especially if it is a solid line indicating that the points on the line are included in the solution (if your inequality has an equal sign).