Question: Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. −5x² − 5x > 0 Use the inequality in the form $f(x) > 0$, to write the intervals determined by the boundary points as they appear from left to right on a number line. Interval Sign (Simplify your answers. Type your answers in interval notation. Use ascending order.)

1. **State the problem:** Solve the inequality $$-5x^2 - 5x > 0$$ and express the solution set in interval notation. 2. **Rewrite the inequality:** Factor out the common factor $$-5x$$: $$-5x^2 - 5x > 0 \implies -5x(x + 1) > 0$$ 3. **Divide both sides by $$-5$$:** Since dividing by a negative number reverses the inequality sign, $$x(x + 1) < 0$$ 4. **Find the boundary points:** Set the expression equal to zero: $$x(x + 1) = 0 \implies x = 0 \text{ or } x = -1$$ These points divide the number line into three intervals: $$(-\infty, -1), (-1, 0), (0, \infty)$$ 5. **Test each interval:** Choose a test point in each interval and evaluate the sign of $$x(x+1)$$. - For $$(-\infty, -1)$$, choose $$x = -2$$: $$(-2)(-2 + 1) = (-2)(-1) = 2 > 0$$ (positive) - For $$(-1, 0)$$, choose $$x = -0.5$$: $$(-0.5)(-0.5 + 1) = (-0.5)(0.5) = -0.25 < 0$$ (negative) - For $$(0, \infty)$$, choose $$x = 1$$: $$(1)(1 + 1) = (1)(2) = 2 > 0$$ (positive) 6. **Determine where $$x(x+1) < 0$$:** This is true only on the interval $$(-1, 0)$$. 7. **Write the solution set:** $$\boxed{(-1, 0)}$$ **Summary:** The solution to $$-5x^2 - 5x > 0$$ is $$x \in (-1, 0)$$. **Interval and sign table:** | Interval | Sign of $$-5x^2 - 5x$$ | |----------------|------------------------| | $$(-\infty, -1)$$ | Negative (since original inequality is $$>0$$, this interval does not satisfy) | | $$(-1, 0)$$ | Positive (solution interval) | | $$(0, \infty)$$ | Negative | Note: The original inequality is $$-5x^2 - 5x > 0$$, which corresponds to $$x(x+1) < 0$$ after dividing by $$-5$$.