In this paper, we solve the additive functional inequality\n\\begin{eqnarray}\n\\|f(x+y)-f(x)-f(y)\\| &\\le & \\left\\|f\\left(\\frac{x+y}{2}\\right)-\\frac{1}{2} f(x) - \\frac{1}{2} f(y)\\right\\| \\end{eqnarray}\nand the quadratic functional inequality\n\\begin{eqnarray}\n&& \\|f(x+y)+f(x-y)-2f(x)-2f(y)\\| \\\\ && \\qquad \\le \\left\\|f\\left(\\frac{x+y}{2}\\right) + f\\left(\\frac{x-y}{2}\\right)-\\frac{1}{2} f(x) - \\frac{1}{2} f(y)\\right\\| . \\nonumber\n\\end{eqnarray}\n\nFurthermore, we prove the Hyers-Ulam stability of the functional inequalities (0.1) and (0.2) in Banach spaces and prove the Hyers-Ulam stability of functional equations associated with the functional inequalities (0.1) and (0.2) in Banach spaces.