1. **State the problem:** Solve the inequality $ (3 - x)(5x + 8) \geq 9 - 3x $. 2. **Expand the left side:** Use distributive property: $$ (3 - x)(5x + 8) = 3 \cdot 5x + 3 \cdot 8 - x \cdot 5x - x \cdot 8 = 15x + 24 - 5x^2 - 8x $$ Simplify: $$ 15x + 24 - 5x^2 - 8x = -5x^2 + 7x + 24 $$ 3. **Rewrite the inequality:** $$ -5x^2 + 7x + 24 \geq 9 - 3x $$ 4. **Bring all terms to one side:** $$ -5x^2 + 7x + 24 - 9 + 3x \geq 0 $$ Simplify: $$ -5x^2 + 10x + 15 \geq 0 $$ 5. **Multiply both sides by -1 to make the quadratic positive, remembering to flip the inequality sign:** $$ 5x^2 - 10x - 15 \leq 0 $$ 6. **Divide entire inequality by 5:** $$ x^2 - 2x - 3 \leq 0 $$ 7. **Factor the quadratic:** $$ (x - 3)(x + 1) \leq 0 $$ 8. **Find critical points:** $$ x = 3 \quad \text{and} \quad x = -1 $$ 9. **Determine intervals to test:** - Interval 1: $(-\infty, -1)$ - Interval 2: $[-1, 3]$ - Interval 3: $(3, \infty)$ 10. **Test values in each interval:** - For $x = -2$ in Interval 1: $$ (-2 - 3)(-2 + 1) = (-5)(-1) = 5 > 0 $$ (Does not satisfy $\leq 0$) - For $x = 0$ in Interval 2: $$ (0 - 3)(0 + 1) = (-3)(1) = -3 \leq 0 $$ (Satisfies inequality) - For $x = 4$ in Interval 3: $$ (4 - 3)(4 + 1) = (1)(5) = 5 > 0 $$ (Does not satisfy) 11. **Conclusion:** The solution to the inequality is $$ \boxed{[-1, 3]} $$ This means all $x$ values between $-1$ and $3$, inclusive, satisfy the original inequality.