a) x2 - 2x -8 <0;
To solve the inequality \( x^2 - 2x - 8 < 0 \), follow these steps:
Factor the quadratic expression: We need to factor \( x^2 - 2x - 8 \). We are looking for two numbers that multiply to \(-8\) (the constant term) and add to \(-2\) (the coefficient of \(x\)). These numbers are \( -4 \) and \( 2 \).
So we can factor the expression as: \[ (x - 4)(x + 2) < 0 \]
Find the critical points: Set the factored form equal to zero to find the critical points: \[ (x - 4)(x + 2) = 0 \] This gives us: \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
The critical points are \( x = -2 \) and \( x = 4 \).
Test intervals around the critical points: We will test the sign of the expression in the intervals determined by the critical points:
Interval 1: \( (-\infty, -2) \)
Interval 2: \( (-2, 4) \)
Interval 3: \( (4, \infty) \)
For \( x < -2 \) (choose \( x = -3 \)): \[ (-3 - 4)(-3 + 2) = (-7)(-1) = 7 > 0 \]
For \( -2 < x < 4 \) (choose \( x = 0 \)): \[ (0 - 4)(0 + 2) = (-4)(2) = -8 < 0 \]
For \( x > 4 \) (choose \( x = 5 \)): \[ (5 - 4)(5 + 2) = (1)(7) = 7 > 0 \]
Determine the solution based on the intervals: The expression \( (x - 4)(x + 2) < 0 \) holds true in the interval: \[ (-2, 4) \]
Final solution: Since the inequality is strict (it is < 0 and does not include equal to), we do not include the points -2 and 4.
Thus, the solution to the inequality \( x^2 - 2x - 8 < 0 \) is: \[ x \in (-2, 4) \]
