+121 To solve the inequality \(x(x + 6) > 16\), follow these steps:
1. Expand the left side of the inequality:
\(x^2 + 6x > 16\)
2. Move all terms to one side of the inequality:
\(x^2 + 6x - 16 > 0\)
3. Factor the quadratic expression:
\((x + 8)(x - 2) < 0\)
Now we have the factors of the quadratic expression. To determine the intervals where the inequality is satisfied, you can use a sign chart or test points within the intervals. The inequality is satisfied when the expression is greater than 0. Let's analyze the sign of the expression within different intervals:
- When \(x < -8\):
Both factors \((x + 8)\) and \((x - 2)\) are negative. Their product is positive. So, the inequality holds in this interval.
- When \(-8 < x < 2\):
\((x + 8)\) is positive, and \((x - 2)\) is negative. Their product is negative. So, the inequality holds in this interval.
- When \(x > 2\):
Both factors \((x + 8)\) and \((x - 2)\) are positive. Their product is negative. So, the inequality does not hold in this interval.
Now, we can write the solution in interval notation:
Solution: \((- \infty, -8) \cup (-8,2)\)
This indicates that the inequality \(x(x + 6) > 16\) is satisfied when \(x\) is in the intervals from negative infinity to -8 (excluding -8), and from -8 (excluding -8) to 2 (excluding 2).
