Question

Use a graphing calculator to graphically solve the radical equation. Check the solution algebraically. $$\sqrt{3 x-2}=4-x$$

Short answer

The short answer will depend upon the exact intersection point one figures out from the graph. It has to be confirmed algebraically by substituting the found x-value in the original equation and checking if both sides are equal.

Step-by-step solution

Graph the Functions

The radical equation is actually given by two different functions: \(f(x)=\sqrt{3 x-2}\) and \(g(x)=4-x\). Graph these functions using a graphing calculator.

Interpret the Graph

Look at the graph, the solution to the original radical equation we're looking for is the x-value where \(f(x)\) and \(g(x)\) intersect. This point of intersection shows where both equations hold true, which is what you need for the solution of the equation.

Check Solution Algebraically

Once you've located the x-value where \(f(x)\) and \(g(x)\) intersect, this is your solution. But to be sure, plug this x-value back into your original equation to verify that both sides are equal, hence confirming the solution.

Key concepts

Graphing Calculator

To solve radical equations, a graphing calculator can be an invaluable tool. These powerful devices enable students to visualize complex algebraic expressions and understand how different functions behave.

When dealing with the given problem \(\sqrt{3x - 2} = 4 - x\), the first step is to input the two expressions \(\sqrt{3x - 2}\) and \(4 - x\) into the graphing calculator separately. Modern graphing calculators have dedicated functions that allow for the graphing of square roots and other radicals. It's important to familiarize yourself with your calculator's specific functions, as this will make inputting the expressions more straightforward.

In addition, these calculators often come with zoom and trace features. These features help in accurately finding the points of intersection between two curves, which is critical in identifying the solution set for the equation. With the proper use of a graphing calculator, complex radical equations become much less intimidating and much more manageable.

Graphical Solution

The graphical solution to an equation offers a visual representation of where the answer lies. In the case of \(\sqrt{3x - 2} = 4 - x\), graphing the functions on either side of the equation gives the visual clues needed to find the solution.

After plotting both \(f(x) = \sqrt{3x - 2}\) and \(g(x) = 4 - x\) on the graphing calculator, the next step is to identify the point of intersection. This point, where both graphs meet, represents the 'x-value' that satisfies both equations. In other words, it's where the original radical equation holds true.

A clear understanding of graph shapes and their intersections is essential. For instance, knowing that a radical function will generally resemble a half-parabola can help in anticipating how the graph of \(f(x)\) might look and how it might intersect with the linear graph of \(g(x)\). Spotting the intersection gives a preliminary 'visual' confirmation of the solution, prior to more precise algebraic verification.

Algebraic Verification

While graphical solutions provide a visual indication of the solution to a radical equation, algebraic verification is crucial to confirm that the solution is correct.

After graphically identifying the x-value that appears to solve the equation \(\sqrt{3x - 2} = 4 - x\), you must substitute this value back into the original equation to check both sides equal one another. This step ensures that your solution is not a graphical anomaly, but a true mathematical solution that balances the equation.

For a thorough verification, solve the radical equation algebraically from the beginning as well. Isolate the radical on one side, square both sides to eliminate the radical, and then solve for 'x'. This process helps reinforce the concept and reveals the underlying algebraic properties of the equation. By aligning both the graphical and algebraic solutions, you solidify your understanding and guarantee the accuracy of your answer.