X2-7x+6<0
To solve the inequality \( x^2 - 7x + 6 < 0 \), we will first find the roots of the corresponding equation:
\[ x^2 - 7x + 6 = 0. \]
We can factor this quadratic equation:
\[ (x - 1)(x - 6) = 0. \]
Setting each factor equal to zero gives us the roots:
- \( x - 1 = 0 \) → \( x = 1 \)
- \( x - 6 = 0 \) → \( x = 6 \)
These roots divide the x-axis into three intervals:
- \( (-\infty, 1) \)
- \( (1, 6) \)
- \( (6, \infty) \)
Next, we need to test each interval to determine where the expression \( x^2 - 7x + 6 \) is less than 0.
Interval \( (-\infty, 1) \): Choose \( x = 0 \): \[ 0^2 - 7(0) + 6 = 6 > 0 \] (Not part of the solution)
Interval \( (1, 6) \): Choose \( x = 2 \): \[ 2^2 - 7(2) + 6 = 4 - 14 + 6 = -4 < 0 \] (Part of the solution)
Interval \( (6, \infty) \): Choose \( x = 7 \): \[ 7^2 - 7(7) + 6 = 49 - 49 + 6 = 6 > 0 \] (Not part of the solution)
Since we are looking for where the expression is less than 0, we find that:
The solution to the inequality \( x^2 - 7x + 6 < 0 \) is:
\[ (1, 6) \]
In interval notation, the solution is:
Final Answer: \( (1, 6) \).
