https://file.aiquickdraw.com/mathbot/file/1728067507313tqrurdnm.jpg

The inequality in the image appears to be:

\[
x + \sqrt{2x - 4} \leq 0
\]

### Steps to solve the inequality:

1. **Domain Restrictions:**
   The square root expression \(\sqrt{2x - 4}\) is only defined when the expression inside the square root is non-negative. So, first, we must solve:
   \[
   2x - 4 \geq 0
   \]
   This simplifies to:
   \[
   x \geq 2
   \]
   Therefore, \(x\) must be greater than or equal to 2 for the square root to be defined.

2. **Isolate the square root term:**
   To solve the inequality \(x + \sqrt{2x - 4} \leq 0\), isolate the square root term on one side:
   \[
   \sqrt{2x - 4} \leq -x
   \]

   However, since the square root is always non-negative (it can only be zero or positive), and \(-x\) is non-positive when \(x \geq 2\), for the inequality to hold, both sides must be zero. Hence:
   \[
   \sqrt{2x - 4} = -x
   \]

   For this to be true, we set:
   \[
   \sqrt{2x - 4} = 0
   \]

3. **Solve for \(x\):**
   From \(\sqrt{2x - 4} = 0\), square both sides:
   \[
   2x - 4 = 0
   \]
   Solving for \(x\), we get:
   \[
   x = 2
   \]

### Conclusion:
The solution to the inequality is \(x = 2\). There are no other values of \(x\) that satisfy the inequality.

Would you like more details or have any further questions? Here are some related topics to consider:

1. What happens if we replace the square root with another function?
2. Can inequalities involving square roots have more than one solution?
3. How does squaring both sides affect inequalities?
4. What happens when dealing with absolute values instead of square roots?
5. What if the inequality was strict (i.e., \(<\) instead of \(\leq\))?

**Tip**: Always check the domain of the function when dealing with square roots or logarithms in inequalities.

Solve the inequality x + √(2x - 4) ≤ 0.

This problem involves solving the inequality x + √(2x - 4) ≤ 0. The solution explores the domain restrictions of square roots and step-by-step isolation of variables. Suitable for students in Grades 9-11, the problem illustrates key algebraic concepts and inequality manipulation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation