Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Use analytic or graphical methods to solve the inequality. $$\sqrt{3 x-5} \leq 4$$

Step-by-Step Solution

Answer

\( \frac{5}{3} \leq x \leq 7 \)

Step 1: Understand the Inequality

Our task is to solve the inequality \( \sqrt{3x - 5} \leq 4 \). This means we want the values of \( x \) for which the expression under the square root, when simplified, results in a value less than or equal to 4.

Step 2: Remove the Square Root

To remove the square root, square both sides of the inequality. This gives us: \( (\sqrt{3x - 5})^2 \leq 4^2 \), simplifying to \( 3x - 5 \leq 16 \).

Step 3: Solve for x

We now solve the simplified inequality \( 3x - 5 \leq 16 \). Add 5 to both sides to get \( 3x \leq 21 \). Next, divide both sides by 3 to isolate \( x \): \( x \leq 7 \).

Step 4: Consider the Domain of the Original Expression

To avoid complex numbers, ensure the expression under the square root is non-negative: \( 3x - 5 \geq 0 \). Solving \( 3x - 5 \geq 0 \) gives \( x \geq \frac{5}{3} \).

Step 5: Combine Results

The solution to the inequality is the intersection of the solutions from Step 3 and Step 4. Thus, \( \frac{5}{3} \leq x \leq 7 \).

Step 6: Graphical Verification (Optional)

Graph the function \( y = \sqrt{3x - 5} \) and the line \( y = 4 \). Verify that the graph of the function lies below the line from \( x = \frac{5}{3} \) to \( x = 7 \) on the x-axis.

Key Concepts

Square Roots

Square roots are fundamental in mathematics and involve finding a number which, when multiplied by itself, yields the original number under the square root sign. In our exercise, the square root is part of the expression \( \sqrt{3x - 5} \). Solving inequalities that include square roots requires understanding their properties, especially when dealing with negative values.

When solving \( \sqrt{3x - 5} \leq 4 \), we need to consider the domain. All values under the square root must be non-negative; otherwise, they become complex numbers, which aren't considered in this problem. Ensure to add the condition \( 3x - 5 \geq 0 \) to work only with permissible values of \( x \).

• Square both sides carefully – it can change the inequality's nature.
• Confirm that your solution respects the domain of the square root expression.

Graphical Methods

Using graphical methods to solve inequalities can be a powerful visual tool. It involves plotting the equations to determine where they satisfy the given inequality. In this exercise, you can graph two elements:

• The function \( y = \sqrt{3x - 5} \)
• The line \( y = 4 \)

Here, you observe where the graph of \( y = \sqrt{3x - 5} \) lies below or on the line \( y = 4 \). Focus on the portion of the graph where \( x \) is plausible – between \( x = \frac{5}{3} \) and \( x = 7 \). At these points, the condition \( \sqrt{3x - 5} \leq 4 \) holds true. Visual confirmation allows you to verify the analytic result and better understand how inequalities behave visually.

Analytic Methods

Analytic methods involve solving inequalities using algebraic manipulation. It is a step-by-step process of transforming the inequality into a simpler form that can be easily interpreted.

Let's walk through the approach used in the exercise:

• Start by isolating the square root: \( \sqrt{3x - 5} \leq 4 \).
• Square both sides to remove the root: \( 3x - 5 \leq 16 \).
• Solve the resulting linear inequality: add 5 resulting in \( 3x \leq 21 \), then divide by 3 to get \( x \leq 7 \).

Additionally, consider the restriction \( 3x - 5 \geq 0 \) to keep \( 3x - 5 \) valid for square roots. Solving this gives \( x \geq \frac{5}{3} \). Finally, combine the results to find the range \( \frac{5}{3} \leq x \leq 7 \), providing a complete solution set. This method ensures a thorough logical sequence and checks for validity at each step.