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Magazine Chapter 5: Problem 52
Use analytic or graphical methods to solve the inequality. $$\sqrt{3 x-5} \leq 4$$
Short Answer
Expert verified
\( \frac{5}{3} \leq x \leq 7 \)
Step by step solution
01
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.
02
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 \( 3x - 5 \leq 16 \).
03
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 \( x \): \( x \leq 7 \).
04
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 \( x \geq \frac{5}{3} \).
05
Combine Results
The solution to the inequality is the intersection of the solutions from Step 3 and Step 4. Thus, \( \frac{5}{3} \leq x \leq 7 \).
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \( x \).
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 鈥� 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:
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 鈥� 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.
- 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 鈥� 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:
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.
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.
