Finding Values Where -x² - 10x is Positive: Inequality Solution

To solve the inequality $-x^2 - 10x > 0$, we can approach it as follows:

• Step 1: Rewrite the inequality.
  We have the inequality $-x^2 - 10x > 0$. To make factorization easier, we first rewrite it:

$-(x^2 + 10x) > 0$

• Step 2: Change signs.
  Multiply through by $-1$ (which reverses the inequality):

$x^2 + 10x < 0$

• Step 3: Factor the equation.
  Set $x^2 + 10x = 0$ to find critical points:

$x(x + 10) = 0$

• This gives the roots $x = 0$ and $x = -10$.
• Step 4: Determine test intervals on a number line.
  The roots divide the number line into intervals: $(-\infty, -10)$, $(-10, 0)$, and $(0, \infty)$.
• Step 5: Test intervals.
  Choose test points from each interval:
• For $(-10, 0)$, pick $x = -5$:
  $x^2 + 10x = (-5)^2 + 10(-5) = 25 - 50 = -25 < 0$.
• For $(-\infty, -10)$, pick $x = -11$, check:
  $(-11)^2 + 10(-11) = 121 - 110 = 11 > 0$.
• For $(0, \infty)$, pick $x = 1$, check:
  $1^2 + 10(1) = 1 + 10 = 11 > 0$.
• Conclusion:
  The only interval where the inequality $x^2 + 10x < 0$ holds is $(-10, 0)$.

Therefore, the solution to the inequality is $-10 < x < 0$.