1. **State the problem:** We need to find all values of $x$ such that $f(x) = g(x)$, where $$f(x) = 0.25x^3 - 2x^2 + x + 7.6$$ $$g(x) = |0.3x| + 2.7$$ 2. **Set the equation:** $$0.25x^3 - 2x^2 + x + 7.6 = |0.3x| + 2.7$$ 3. **Consider the absolute value definition:** - For $x \geq 0$, $|0.3x| = 0.3x$ - For $x < 0$, $|0.3x| = -0.3x$ 4. **Solve for $x \geq 0$:** $$0.25x^3 - 2x^2 + x + 7.6 = 0.3x + 2.7$$ Bring all terms to one side: $$0.25x^3 - 2x^2 + x - 0.3x + 7.6 - 2.7 = 0$$ Simplify: $$0.25x^3 - 2x^2 + 0.7x + 4.9 = 0$$ 5. **Solve for $x < 0$:** $$0.25x^3 - 2x^2 + x + 7.6 = -0.3x + 2.7$$ Bring all terms to one side: $$0.25x^3 - 2x^2 + x + 0.3x + 7.6 - 2.7 = 0$$ Simplify: $$0.25x^3 - 2x^2 + 1.3x + 4.9 = 0$$ 6. **Find roots numerically:** - For $x \geq 0$, solve $$0.25x^3 - 2x^2 + 0.7x + 4.9 = 0$$ - For $x < 0$, solve $$0.25x^3 - 2x^2 + 1.3x + 4.9 = 0$$ Using numerical methods (e.g., graphing or root-finding), approximate solutions to the nearest hundredth: - For $x \geq 0$, root near $x \approx 2.88$ - For $x < 0$, root near $x \approx -1.88$ 7. **Final solutions:** $$x \approx -1.88, \quad x \approx 2.88$$