Problem 58 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \geq 3^{x}\\\ &y \geq 2 \end{aligned}$$

Step-by-Step Solution

Answer

Graph a region above both \( y = 3^x \) and \( y = 2 \).

Step 1: Understand the Inequalities

We have two inequalities to graph: \( y \geq 3^x \) and \( y \geq 2 \). The first inequality involves an exponential function, while the second is a horizontal line.

Step 2: Graph the Exponential Inequality

Consider the equation \( y = 3^x \). This graph is an exponential curve increasing from left to right. For \( y \geq 3^x \), shade the region above this curve, as the inequality indicates values of \( y \) that are greater than or equal to \( 3^x \).

Step 3: Graph the Horizontal Line

Graph the line \( y = 2 \). This is a horizontal line where every point on the line has a y-coordinate of 2. For \( y \geq 2 \), shade the region above this line.

Step 4: Identify the Solution Set

To find the solution set, identify the region that satisfies both inequalities. This is the intersection of the shaded regions from Steps 2 and 3. The solution set is the region above both the curve \( y = 3^x \) and the line \( y = 2 \).

Step 5: Sketch the Graph

Draw the x and y axes. Sketch the exponential curve for \( y = 3^x \), starting at \( (0,1) \) and increasing exponentially. Then draw the horizontal line at \( y = 2 \). Shade the region above both the curve and the line, which is the area representing the solution set.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where the variable is in the exponent. It is often expressed in the form \( y = a^x \), where \( a \) is a constant and \( a > 0 \). In our case, the function is \( y = 3^x \).

• This function represents a curve that rises very quickly. As the value of \( x \) increases, the value of \( y \) increases exponentially.
• The graph of \( y = 3^x \) starts off slow for negative or small values of \( x \) but accelerates as \( x \) becomes larger.
• For \( y \geq 3^x \), we consider all y-values that are at or above the exponential curve.

In graphing an exponential function, it's important to note its behavior near zero. Typically, these functions pass through the point where the base \( a \) is to the power of zero, such as \( (0,1) \) in this example. This function grows without bound but is always above zero.

Solution Set

Identifying the solution set of a system of inequalities involves finding the values that satisfy all inequalities within the system. In this exercise, we have two inequalities, \( y \geq 3^x \) and \( y \geq 2 \).

• Firstly, we determine the area where \( y \) values are greater than or equal to \( 3^x \). This includes the region above the exponential curve.


• Secondly, we inspect the area where \( y \) is greater than or equal to 2. This involves all points above the horizontal line \( y=2 \).

The solution set is where both conditions are true simultaneously. Therefore, the solution set is the region that lies above both the exponential curve and the horizontal line. In graphical terms, it is the overlapping shaded area from both these constraints.

Graphing Techniques

When graphing inequalities, several techniques are helpful in ensuring accuracy. Here, you're dealing with two very different types of graphs: an exponential curve and a horizontal line.

• Start with the exponential graph. Plot key points and remember that the curve should get steeper as \( x \) increases.


• Next, plot the horizontal line. Keep in mind that no matter the value of \( x \), \( y \) remains constant along this line.


• Shade areas correctly. For \( y \geq 3^x \), shade above the curve. For \( y \geq 2 \), shade above this line.

Finally, look for where these two shaded regions intersect. The intersected area represents all points that satisfy both inequalities, showcasing where the solution lies within the coordinate plane.

Inequalities

Understanding inequalities is crucial when solving problems where solutions are ranges of values instead of exact numbers. An inequality such as \( y \geq 3^x \) tells us that y-values that satisfy this inequality lie on or above the curve.

• Inequalities show a relationship where one side is not strictly equal to the other, allowing for a range of possible values.


• The graph of an inequality often involves shading, which visually represents all possible solutions.

In this context, understanding both \( \geq \) ("greater than or equal to") and \( \leq \) ("less than or equal to") is essential. The shading helps us visualize inequalities, indicating regions where all conditions are true. By shading on graphs, you convey the solution set powerfully, showing all values that meet the inequality criteria.