Problem 12 - Thinking Mathematically

Question

Graph each linear inequality. \(y>\frac{1}{4} x\)

Step-by-Step Solution

Answer

To graph the inequality \(y > \frac{1}{4} x\), plot the line for \(y = \frac{1}{4} x\) and then shade the area above the line.

Step 1 - Understanding the inequality

First, it is important to understand the meaning of \(y > \frac{1}{4} x\). This inequality says that y is larger than one fourth of x.

Step 2 - Plotting the Line

Next thing to do is to plot the line as if it were \(y = \frac{1}{4} x\). Start at the origin (0,0) and then for every 4 units x moves to the right, move one unit up for y. This will give you the line for the equation

Step 3 - Shading the Area

Since this is an inequality, we need to shade an area of the graph. Because the inequality is \(y > \frac{1}{4} x\), we shade the area ABOVE the line. The shaded area represents all the points (x, y) that satisfy the inequality.

Key Concepts

Inequality Representation

Understanding inequality representation is essential in grasping how to graph inequalities. In the world of algebra, an inequality reveals the relationship between two expressions that are not necessarily equal, indicated by symbols like '<', '>', '\(\leq\)', and '\(\geq\)'. In our given exercise, the inequality \(y > \frac{1}{4} x\) denotes that the value of \(y\) is strictly greater than \(\frac{1}{4}\) times the value of \(x\). Notice the lack of an equal sign; it's crucial, as this means that the line \(y = \frac{1}{4} x\) itself is not a part of the set of solutions. Rather, it acts as a boundary that helps us see which side of the line corresponds to the solutions. Knowing the difference between '\(=\)' and '>' is pivotal in determining the approach to take when you're plotting the graph.

Plotting Inequalities

When it comes to plotting inequalities, the process begins similarly to graphing a linear equation. You plot the boundary line, which is done by translating the inequality into an equation. For instance, \(y > \frac{1}{4} x\) becomes \(y = \frac{1}{4} x\) for the purpose of plotting. Once the boundary is set, a dashed line is often used to indicate that the line itself is not included in the solution set for a '>' or '<' inequality. If the inequality includes a '\(\geq\)' or '\(\leq\)', then you would use a solid line. After plotting the line, the next step is determining the region where the inequality holds true. This is where you'll prepare to shade the graph to visually display the solution set for the inequality.

Choosing the Correct Half-Plane

To choose the correct half-plane, pick a test point not on the boundary line (often the origin if it isn't on the line) and substitute it into the inequality. If the inequality holds true, the side that contains the test point is the solution region; otherwise, it's the opposite side.

Shading Graph Regions

Shading graph regions is how we visually make sense of the solution set for an inequality. Once the boundary line is drawn and a test point is used to determine which side of the line matches the inequality, the next step is to shade the correct region. Shading is quite intuitive: if the inequality is '\(y > something\)', you shade above the boundary line; if it is '\(y < something\)', below the line is shaded. The shaded area on the graph represents all the points that satisfy the inequality. It’s critical to recognize that every point in that shaded region is a solution, meaning there are infinitely many solutions to an inequality. By shading the graph, we communicate a great deal of information rather succinctly, providing a quick way to see which values of \(x\) and \(y\) will make the inequality true.

Linear Equation Graphing

In the context of our inequality problem, graphing the corresponding linear equation serves as the starting point for graphing the entire inequality. The linear equation that forms the boundary for our inequality \(y > \frac{1}{4} x\) is \(y = \frac{1}{4} x\). Graphing it involves plotting two or more points that satisfy the equation and then drawing a straight line through them. The slope, represented by the coefficient of \(x\) in the equation, dictates how steep the line is. Here, a slope of \(\frac{1}{4}\) means for every 4 units \(x\) increases, \(y\) increases by one unit. It's particularly useful to remember to plot the y-intercept as one of the points, which, in this case, is the origin (0, 0) since no \(y\)-intercept is stipulated. After identifying the slope and intercept, you can sketch the line smoothly with a ruler. The precision of your line will directly affect the accuracy of your inequality shading, making this step quite important for the overall problem's solution.