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# Solve the inequality $x(x + 6) > 16.$ Write your answer in interval notation

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Aug 16, 2023

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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:
$$x^2 + 6x - 16 > 0$$

$$(x + 8)(x - 2) < 0$$

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).

Aug 17, 2023