For which values of x is the following inequality true : `x^2 < 2x`.

Options

A. `x lt 0`
B. `0 lt x lt 2`
C. `-2 lt x lt 2`
D. `x lt 2`

Answer

To solve the inequality \( x^2 < 2x \), we can follow these steps:

### Step 1: Rearrange the Inequality
First, we will rearrange the inequality to bring all terms to one side:
\[ x^2 - 2x < 0 \]

### Step 2: Factor the Expression
Next, we can factor the left-hand side:
\[ x(x - 2) < 0 \]

### Step 3: Find Critical Points
Now, we need to find the critical points where the expression equals zero. We set the factored expression equal to zero:
\[ x(x - 2) = 0 \]
This gives us the critical points:
\[ x = 0 \quad \text{and} \quad x = 2 \]

### Step 4: Test Intervals
We will test the intervals determined by the critical points \( x = 0 \) and \( x = 2 \). The intervals to test are:
1. \( (-\infty, 0) \)
2. \( (0, 2) \)
3. \( (2, \infty) \)

#### Test Interval 1: \( (-\infty, 0) \)
Choose a test point, for example, \( x = -1 \):
\[ (-1)(-1 - 2) = (-1)(-3) = 3 \quad (\text{not less than } 0) \]
So, this interval does not satisfy the inequality.

#### Test Interval 2: \( (0, 2) \)
Choose a test point, for example, \( x = 1 \):
\[ (1)(1 - 2) = (1)(-1) = -1 \quad (\text{less than } 0) \]
So, this interval satisfies the inequality.

#### Test Interval 3: \( (2, \infty) \)
Choose a test point, for example, \( x = 3 \):
\[ (3)(3 - 2) = (3)(1) = 3 \quad (\text{not less than } 0) \]
So, this interval does not satisfy the inequality.

### Step 5: Conclusion
The inequality \( x^2 < 2x \) is satisfied for values of \( x \) in the interval:
\[ (0, 2) \]

Thus, the solution to the inequality is:
\[ 0 < x < 2 \]