Zero Solutions 13Cb89

1. **State the problem:** We are given the equation $$h(-3x - 9) = -9(x + 9)$$ and need to find the value of $$h$$ for which the equation has zero solutions. 2. **Rewrite the equation:** $$h(-3x - 9) = -9(x + 9)$$ 3. **Distribute both sides:** $$-3hx - 9h = -9x - 81$$ 4. **Group like terms:** Bring all terms involving $$x$$ to one side and constants to the other: $$-3hx + 9x = -81 + 9h$$ 5. **Factor out $$x$$ on the left side:** $$x(-3h + 9) = -81 + 9h$$ 6. **Analyze the equation for zero solutions:** - If the coefficient of $$x$$ is zero, i.e., $$-3h + 9 = 0$$, then the equation becomes: $$0 imes x = -81 + 9h$$ - This simplifies to: $$0 = -81 + 9h$$ 7. **Solve for $$h$$ when coefficient of $$x$$ is zero:** $$-3h + 9 = 0 \implies -3h = -9 \implies h = 3$$ 8. **Check the constant term when $$h=3$$:** $$-81 + 9(3) = -81 + 27 = -54 \neq 0$$ 9. **Interpretation:** - When $$h=3$$, the equation reduces to $$0 = -54$$, which is false. - This means there are **no solutions** because the equation is inconsistent. 10. **Answer the blanks:** - This equation will have zero solutions when $$h = 3$$ because you get zero solutions when you have **zero** number of $$x$$'s on either side of the equation and the constants are **not equal**.