Make the following expressions free from modulus sign :
(i) `|x^(2) -3x-4|` , (ii) `|x^(2)-7x+10|` if `x lt 5`

Answer

To make the given expressions free from modulus signs, we will analyze the expressions based on their roots and the intervals they create.

### (i) Expression: `|x^2 - 3x - 4|`

1. **Find the roots of the quadratic equation**:
\[ x^2 - 3x - 4 = 0 \]
Factoring gives us:
\[ (x + 1)(x - 4) = 0 \]
Thus, the roots are \(x = -1\) and \(x = 4\).

2. **Determine intervals based on the roots**:
The roots divide the number line into three intervals:
- \( (-\infty, -1) \)
- \( (-1, 4) \)
- \( (4, \infty) \)

3. **Test the sign of the expression in each interval**:
- For \(x < -1\) (e.g., \(x = -2\)):
\[ (-2)^2 - 3(-2) - 4 = 4 + 6 - 4 = 6 \quad (\text{positive}) \]
- For \(-1 < x < 4\) (e.g., \(x = 0\)):
\[ 0^2 - 3(0) - 4 = -4 \quad (\text{negative}) \]
- For \(x > 4\) (e.g., \(x = 5\)):
\[ 5^2 - 3(5) - 4 = 25 - 15 - 4 = 6 \quad (\text{positive}) \]

4. **Write the expression without modulus**:
Since we are considering \(x < 5\), we need to consider two cases:
- For \(x < -1\): The expression is positive, so:
\[ |x^2 - 3x - 4| = x^2 - 3x - 4 \]
- For \(-1 < x < 4\): The expression is negative, so:
\[ |x^2 - 3x - 4| = -(x^2 - 3x - 4) = -x^2 + 3x + 4 \]
- For \(4 < x < 5\): The expression is positive, so:
\[ |x^2 - 3x - 4| = x^2 - 3x - 4 \]

Thus, the expression without modulus is:
\[ |x^2 - 3x - 4| = \begin{cases} x^2 - 3x - 4 & \text{if } x < -1 \\ -x^2 + 3x + 4 & \text{if } -1 < x < 4 \\ x^2 - 3x - 4 & \text{if } 4 < x < 5 \end{cases} \]

### (ii) Expression: `|x^2 - 7x + 10|` if `x < 5`

1. **Find the roots of the quadratic equation**:
\[ x^2 - 7x + 10 = 0 \]
Factoring gives us:
\[ (x - 2)(x - 5) = 0 \]
Thus, the roots are \(x = 2\) and \(x = 5\).

2. **Determine intervals based on the roots**:
The roots divide the number line into three intervals:
- \( (-\infty, 2) \)
- \( (2, 5) \)
- \( (5, \infty) \)

3. **Test the sign of the expression in each interval**:
- For \(x < 2\) (e.g., \(x = 0\)):
\[ 0^2 - 7(0) + 10 = 10 \quad (\text{positive}) \]
- For \(2 < x < 5\) (e.g., \(x = 3\)):
\[ 3^2 - 7(3) + 10 = 9 - 21 + 10 = -2 \quad (\text{negative}) \]

4. **Write the expression without modulus**:
Since we are considering \(x < 5\), we need to consider two cases:
- For \(x < 2\): The expression is positive, so:
\[ |x^2 - 7x + 10| = x^2 - 7x + 10 \]
- For \(2 < x < 5\): The expression is negative, so:
\[ |x^2 - 7x + 10| = -(x^2 - 7x + 10) = -x^2 + 7x - 10 \]

Thus, the expression without modulus is:
\[ |x^2 - 7x + 10| = \begin{cases} x^2 - 7x + 10 & \text{if } x < 2 \\ -x^2 + 7x - 10 & \text{if } 2 < x < 5 \end{cases} \]