Answer:
To solve the inequality \(x^2 - x + 6 < 2(x + 2)\), let's first simplify:
\[ x^2 - x + 6 < 2x + 4 \]
Subtract \(2x + 4\) from both sides:
\[ x^2 - x + 6 - 2x - 4 < 0 \]
Combine like terms:
\[ x^2 - 3x + 2 < 0 \]
Now, let's factor the quadratic expression:
\[ (x - 1)(x - 2) < 0 \]
To find the solution, consider the sign of each factor for different intervals on the number line:
- If \( x < 1 \), both factors are negative, so the expression is positive.
- If \( 1 < x < 2 \), the first factor is positive, and the second factor is negative, so the expression is negative.
- If \( x > 2 \), both factors are positive, so the expression is positive.
Therefore, the solution to the inequality is \( 1 < x < 2 \).
