`|x^(2) - 4x| lt 5`

Options

A. `-1 le x le 5`
B. `1 le x le 5`
C. `-1 le x le 1`
D. `-1 lt x lt 5`

Answer

To solve the inequality \( |x^2 - 4x| < 5 \), we can break it down into two separate inequalities based on the definition of absolute value.

### Step 1: Set up the inequalities
The absolute value inequality \( |A| < B \) can be rewritten as:
\[ -B < A < B \]
In our case, we have:
\[ -5 < x^2 - 4x < 5 \]

### Step 2: Solve the first inequality
Let's solve the first part:
\[ x^2 - 4x > -5 \]
Rearranging gives:
\[ x^2 - 4x + 5 > 0 \]
Now, we can find the roots of the quadratic equation \( x^2 - 4x + 5 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -4, c = 5 \):
\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} \]
\[ x = \frac{4 \pm \sqrt{16 - 20}}{2} \]
\[ x = \frac{4 \pm \sqrt{-4}}{2} \]
Since the discriminant is negative, there are no real roots, and the quadratic \( x^2 - 4x + 5 \) is always positive.

### Step 3: Solve the second inequality
Now, let's solve the second part:
\[ x^2 - 4x < 5 \]
Rearranging gives:
\[ x^2 - 4x - 5 < 0 \]
Now, we can find the roots of the quadratic equation \( x^2 - 4x - 5 = 0 \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -4, c = -5 \):
\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1} \]
\[ x = \frac{4 \pm \sqrt{16 + 20}}{2} \]
\[ x = \frac{4 \pm \sqrt{36}}{2} \]
\[ x = \frac{4 \pm 6}{2} \]
This gives us two roots:
\[ x = \frac{10}{2} = 5 \quad \text{and} \quad x = \frac{-2}{2} = -1 \]

### Step 4: Test intervals
Now we need to test the intervals determined by the roots \(-1\) and \(5\):
1. For \( x < -1 \), choose \( x = -2 \):
\[ (-2)^2 - 4(-2) - 5 = 4 + 8 - 5 = 7 \quad (\text{positive}) \]
2. For \( -1 < x < 5 \), choose \( x = 0 \):
\[ 0^2 - 4(0) - 5 = -5 \quad (\text{negative}) \]
3. For \( x > 5 \), choose \( x = 6 \):
\[ 6^2 - 4(6) - 5 = 36 - 24 - 5 = 7 \quad (\text{positive}) \]

### Step 5: Conclusion
The solution to the inequality \( x^2 - 4x - 5 < 0 \) is:
\[ -1 < x < 5 \]
Thus, the solution to the original inequality \( |x^2 - 4x| < 5 \) is:
\[ x \in (-1, 5) \]