Solve $3^{x+1}=2^x+4$ graphically.
Answer (2)
To solve the equation 3 x + 1 = 2 x + 4 graphically, we can define the function f ( x ) = 3 x + 1 − 2 x − 4 and find its roots. The approximate root is x ≈ 0.545 , indicating that this is the solution to the original equation. Therefore, the solution is x ≈ 0.545 .
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Rewrite the equation as a function: f ( x ) = 3 x + 1 − 2 x − 4 .
Find the roots of the function f ( x ) using the tool.
The approximate real-valued root is x ≈ 0.5449 .
Compare the result with the given possible solutions. The solution is 0.545 .
Explanation
Understanding the Problem We are given the equation 3 x + 1 = 2 x + 4 and asked to solve it graphically. This means we need to find the value(s) of x that satisfy this equation. We are also given a few possible solutions to check.
Defining the Function Let's rewrite the equation as a function: f ( x ) = 3 x + 1 − 2 x − 4 . The solutions to the original equation are the roots (or x-intercepts) of this function, i.e., the values of x for which f ( x ) = 0 .
Graphical Interpretation To solve graphically, we would plot the function f ( x ) and look for the points where the graph crosses the x-axis. However, we don't have a graphing tool here. Instead, we can use the tool to find the approximate real-valued roots of the function.
Finding the Root Using the tool, we find that the approximate real-valued root of the function f ( x ) = 3 x + 1 − 2 x − 4 in the interval [ 0 , 1 ] is x ≈ 0.5449 .
Checking Possible Solutions Comparing this result with the given possible solutions, we see that 0.545 is very close to the calculated root. Therefore, x ≈ 0.545 is a solution to the equation 3 x + 1 = 2 x + 4 .
Final Answer Therefore, the solution to the equation 3 x + 1 = 2 x + 4 is approximately 0.545 .
Examples
Graphical solutions are useful in many areas, such as physics, engineering, and economics, to visualize and understand the behavior of complex equations. For example, in circuit analysis, one might graph the voltage and current relationship to find the operating point of a circuit. In economics, supply and demand curves are graphed to find the equilibrium price and quantity. These graphical methods provide a visual representation of the problem, making it easier to understand and solve.
