Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the inequality. $$y<\ln x$$

Step-by-Step Solution

Answer

The drawn graph starts from point (1,0) and slowly moves upwards to the right as x increases, representing the function \(y=\ln x\). The shaded area is everything below this curve in the xy-plane, which represents where \(y<\ln x\).

Step 1: Graphing the Boundary Line

Plot the function \(y = \ln x\). This is the boundary line of the area that represents the solution to the inequality. Since it is a logarithmic function, it starts from (1,0) and climbs upwards to the right, slowly.

Step 2: Identifying the Inequality Symbol

The inequality symbol in this exercise is '<'. This symbol means that the solution will contain all the values that are less than the boundary line.

Step 3: Shading the Correct Region

Because the inequality symbol is '<', shade the area under the boundary line. The shaded region will contain all points (x, y) where y is less than \(\ln x\).

Key Concepts

Logarithmic Function

When studying the behavior of a logarithmic function, like the one in our example, it's crucial to grasp its characteristics. A logarithmic function is basically the inverse of an exponential function. It is defined for positive real numbers and has the form f(x) = log_b(x), where x is the value, and b is the base of the logarithm, often a positive real number larger than 1.

The graph of the logarithmic function y = \(\ln x\), where \ln denotes the natural logarithm with the base e (approximately 2.718), shows that it passes through the point (1,0), goes infinitely towards the left as it approaches the y-axis, and ascends slowly without bound as x increases towards the right. The curve never touches the y-axis (x=0), which is an asymptote - a line that the curve approaches but never intersects.

Inequality Graphing

The art of inequality graphing is a valuable tool for visualizing the solutions to equations and inequalities. Much like its cousin, the equation graph, an inequality graph translates mathematical expressions into visual areas on a coordinate plane. To graph an inequality, one first plots the corresponding equality - the boundary line.

However, unlike graphing a simple equation, graphing an inequality requires us to consider which side of the boundary line is included in the solution set. The inequality's symbol guides us here. For instance, if the symbol is '<' or '\(\leq\)', we will be focused on the area below the boundary line. If the symbol is '>' or '\(\geq\)', we look above the boundary line. In our case, with y < \ln x, the area beneath the curve of the logarithmic function contains the solution set.

Shading Regions

Once the boundary line is plotted and the inequality symbol analyzed, we move on to shading regions to represent the set of solutions. The process is straightforward but crucial - the shading indicates where the inequality holds true. In our problem, because we're dealing with the inequality y < \ln x, the region shaded is under the curve.

The process of choosing where to shade involves a simple test: pick a point not on the boundary, substitute its coordinates into the inequality and see if you get a true statement. If true, the side containing that test point is the correct region to shade. If false, then shade the opposite side. It's also helpful to remember that if the inequality were a '\(\leq\)' or '\(\geq\)', the boundary line itself would also be shaded, indicating that points on the line are included in the solution.

Boundary Line

The boundary line is the edge between solutions and non-solutions in inequality graphing. For the inequality y < \ln x, the boundary line is the graph of the equation y = \ln x. This line serves as a reference to determine which side of it will be shaded. It's paramount to recognize that this line does not include any points of the solution set when dealing with '<' (less than) or '>' (greater than) inequalities.

Therefore, in such cases, the boundary line remains dashed or dotted to signify that it's not part of the solution. Should the inequality contain '\(\leq\)' or '\(\geq\)', the line would be solid, indicating that points on the line satisfy the inequality. Boundary lines are integral in distinguishing between areas of solutions and non-solutions and should be graphed with care to ensure clarity in understanding the solutions to the inequality.