https://r2.erweima.ai/mathbot/image/1726542659228l4o5m2i2.jpg

The inequality provided in the image is:

\[
\sqrt{\frac{x - 2}{12}} < 4
\]

To solve this inequality, let's proceed step by step.

### Step 1: Eliminate the square root
We can eliminate the square root by squaring both sides of the inequality:

\[
\left( \sqrt{\frac{x - 2}{12}} \right)^2 < 4^2
\]

This simplifies to:

\[
\frac{x - 2}{12} < 16
\]

### Step 2: Eliminate the denominator
Multiply both sides of the inequality by 12 to remove the denominator:

\[
x - 2 < 16 \times 12
\]

\[
x - 2 < 192
\]

### Step 3: Solve for \(x\)
Now, add 2 to both sides to isolate \(x\):

\[
x < 192 + 2
\]

\[
x < 194
\]

### Step 4: Consider the domain
Since we initially dealt with a square root, the expression inside the square root must be non-negative. Therefore:

\[
\frac{x - 2}{12} \geq 0
\]

Multiplying both sides by 12:

\[
x - 2 \geq 0
\]

\[
x \geq 2
\]

### Final Answer:
The solution to the inequality is:

\[
2 \leq x < 194
\]

This is the interval where the inequality holds true.

Would you like further details or have any questions?

### Related Questions:
1. How do you solve inequalities involving absolute values?
2. What are the rules for squaring both sides of an inequality?
3. How do you graph inequalities on a number line?
4. What happens when inequalities involve cube roots instead of square roots?
5. How do we solve inequalities with variables on both sides?

### Tip:
Always remember to check the domain restrictions when solving inequalities involving square roots, as the expression inside the root must be non-negative.

resolver la desigualdad sqrt((x - 2)/12) < 4

Learn how to solve the inequality sqrt((x - 2)/12) < 4 with a step-by-step breakdown. This problem covers essential algebraic concepts, including solving inequalities involving square roots and understanding domain restrictions. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation