X2-7x+6<0
To solve the inequality \( x^2 - 7x + 6 < 0 \), we can start by factoring the quadratic expression.
Factor the quadratic: We need to find two numbers that multiply to \( 6 \) (the constant term) and add to \(-7\) (the coefficient of \( x \)). The numbers \(-1\) and \(-6\) work because: \(-1 \times -6 = 6\) and \(-1 + -6 = -7\).
Thus, we can factor the expression: \[ x^2 - 7x + 6 = (x - 1)(x - 6) \]
Set the inequality: Now we rewrite the inequality as: \[ (x - 1)(x - 6) < 0 \]
Identify the critical points: From the factors, we find the roots: \[ x - 1 = 0 \implies x = 1 \] \[ x - 6 = 0 \implies x = 6 \] So the critical points are \( x = 1 \) and \( x = 6 \).
Test the intervals: We have three intervals to test:
- \( (-\infty, 1) \)
- \( (1, 6) \)
- \( (6, \infty) \)
Let's test a point in each interval:
- For \( x = 0 \) (in \( (-\infty, 1) \)): \[ (0 - 1)(0 - 6) = (-1)(-6) = 6 > 0 \]
- For \( x = 3 \) (in \( (1, 6) \)): \[ (3 - 1)(3 - 6) = (2)(-3) = -6 < 0 \]
- For \( x = 7 \) (in \( (6, \infty) \)): \[ (7 - 1)(7 - 6) = (6)(1) = 6 > 0 \]
Determine where the inequality holds: The product \( (x - 1)(x - 6) < 0 \) is true in the interval \( (1, 6) \).
Write the solution: Therefore, the solution to the inequality \( x^2 - 7x + 6 < 0 \) is: \[ \boxed{(1, 6)} \]
