X2-7x+6<0

To solve the inequality \(x^2 - 7x + 6 < 0\), we first need to find the roots of the corresponding quadratic equation \(x^2 - 7x + 6 = 0\).

Step 1: Factor the quadratic

We can factor the quadratic as follows:

\[ x^2 - 7x + 6 = (x - 1)(x - 6) \]

Step 2: Find the roots

Setting the factors equal to zero gives us the roots:

\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]

Step 3: Determine intervals

The roots divide the number line into three intervals:

1. \( (-\infty, 1) \)
2. \( (1, 6) \)
3. \( (6, \infty) \)

Step 4: Test the sign of the expression

We can test the sign of \( (x - 1)(x - 6) \) in each interval:

1. For \( x < 1 \) (e.g., \( x = 0 \)): \[ (0 - 1)(0 - 6) = (-1)(-6) = 6 \quad (\text{positive}) \]

2. For \( 1 < x < 6 \) (e.g., \( x = 4 \)): \[ (4 - 1)(4 - 6) = (3)(-2) = -6 \quad (\text{negative}) \]

3. For \( x > 6 \) (e.g., \( x = 7 \)): \[ (7 - 1)(7 - 6) = (6)(1) = 6 \quad (\text{positive}) \]

Step 5: Identify where the inequality holds

We want the expression \( (x - 1)(x - 6) < 0 \). From our testing:

• The expression is negative in the interval \( (1, 6) \).

Step 6: Conclusion

Thus, the solution to the inequality \( x^2 - 7x + 6 < 0 \) is:

\[ \boxed{(1, 6)} \]

This means that \( x \) must be greater than 1 and less than 6.