Solve the following inequality. $$ x^{3}+x^{2}

Question

Solve the following inequality. $$ x^{3}+x^{2}<9 x+9 $$ Write your answer as an interval or union of intervals. If there is no real solution, click on "No solution".

UpStudy AI · Accepted Answer

We start with the inequality
$$
x^{3} + x^{2} < 9x + 9.
$$
Subtract $$9x+9$$ from both sides:
$$
x^{3} + x^{2} - 9x - 9 < 0.
$$
Factor by grouping:
$$
x^{3}+x^{2}-9x-9 = \left(x^{3}+x^{2}\right) - \left(9x+9\right) = x^{2}(x+1) - 9(x+1) = (x+1)(x^{2}-9).
$$
Notice that $$x^{2}-9$$ is a difference of squares:
$$
x^{2}-9 = (x+3)(x-3).
$$
Thus, the inequality becomes
$$
(x+1)(x+3)(x-3) < 0.
$$

The critical points are obtained by setting each factor equal to zero:
$$
x+1=0 \Rightarrow x=-1,\quad x+3=0 \Rightarrow x=-3,\quad x-3=0 \Rightarrow x=3.
$$

We test the sign in the intervals determined by these critical points:

1. For $$x < -3$$: Choose $$x = -4$$:
   $$
   x+1 = -3 \, (\text{negative}),\quad x+3 = -1 \, (\text{negative}),\quad x-3 = -7 \, (\text{negative}).
   $$
   The product is $$(-) \times (-) \times (-) = -$$ (negative).

2. For $$-3 < x < -1$$: Choose $$x = -2$$:
   $$
   x+1 = -1 \, (\text{negative}),\quad x+3 = 1 \, (\text{positive}),\quad x-3 = -5 \, (\text{negative}).
   $$
   The product is $$(-) \times (+) \times (-) = +$$ (positive).

3. For $$-1 < x < 3$$: Choose $$x = 0$$:
   $$
   x+1 = 1 \, (\text{positive}),\quad x+3 = 3 \, (\text{positive}),\quad x-3 = -3 \, (\text{negative}).
   $$
   The product is $$(+) \times (+) \times (-) = -$$ (negative).

4. For $$x > 3$$: Choose $$x = 4$$:
   $$
   x+1 = 5 \, (\text{positive}),\quad x+3 = 7 \, (\text{positive}),\quad x-3 = 1 \, (\text{positive}).
   $$
   The product is $$(+) \times (+) \times (+) = +$$ (positive).

We require the product to be negative. This happens in the intervals:
$$
(-\infty, -3) \quad \text{and} \quad (-1, 3).
$$

Thus, the solution to the inequality is:
$$
(-\infty, -3) \cup (-1, 3).
$$
The solution to the inequality $$ x^{3} + x^{2} < 9x + 9 $$ is $$ x $$ in the intervals $$ (-\infty, -3) $$ and $$ (-1, 3) $$.

Solve the following inequality.

Write your answer as an interval or union of intervals.
If there is no real solution, click on “No solution”.

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Solve the following inequality. Write your answer as an interval or union of intervals. If there is no real solution, click on “No solution”.