determine range of values of x for which x(x-2) </ 2x-4

We are tasked with finding the range of \( x \) that satisfies the inequality:

\[
x(x - 2) < 2x - 4
\]

### Step-by-step solution:

1. **Expand both sides:**
   On the left-hand side, distribute \( x \):
   \[
   x(x - 2) = x^2 - 2x
   \]
   Now the inequality becomes:
   \[
   x^2 - 2x < 2x - 4
   \]

2. **Move all terms to one side:**
   Subtract \( 2x - 4 \) from both sides:
   \[
   x^2 - 2x - (2x - 4) < 0
   \]
   Simplify:
   \[
   x^2 - 4x + 4 < 0
   \]

3. **Factor the quadratic expression:**
   The left-hand side is a perfect square:
   \[
   x^2 - 4x + 4 = (x - 2)^2
   \]
   So the inequality becomes:
   \[
   (x - 2)^2 < 0
   \]

4. **Analyze the inequality:**
   A square of a real number is always non-negative, i.e., \( (x - 2)^2 \geq 0 \). Therefore, \( (x - 2)^2 < 0 \) has **no real solution** because no real number squared can be strictly less than zero.

### Conclusion:
There is **no real value** of \( x \) that satisfies the inequality \( x(x - 2) < 2x - 4 \).

---

Would you like more details or have any questions?

Here are 5 related questions you might find useful:

1. How do we solve quadratic inequalities in general?
2. What are the steps to solve quadratic equations?
3. How do you determine when a quadratic inequality has no real solutions?
4. What is the difference between strict and non-strict inequalities in mathematics?
5. How do you solve inequalities involving higher-degree polynomials?

**Tip:** Always check the discriminant of a quadratic expression to understand the nature of its roots.

Determine the range of values of x for which x(x-2) < 2x-4

Explore the solution to the quadratic inequality x(x - 2) < 2x - 4. This step-by-step guide involves expanding terms, factoring the quadratic expression, and analyzing the nature of squares. Perfect for students in Grades 9-11 studying inequalities and quadratic equations.

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