Function Intersection 669Ff7

1. **State the problem:** We need to find the value(s) of $x$ such that $f(x) = g(x)$ where $f(x) = \frac{x-2}{-2}$ and $g(x) = -4x + 24$. 2. **Set the functions equal:** To find $x$ where $f(x) = g(x)$, solve: $$\frac{x-2}{-2} = -4x + 24$$ 3. **Multiply both sides by $-2$ to clear the denominator:** $$x - 2 = (-2)(-4x + 24)$$ $$x - 2 = 8x - 48$$ 4. **Bring all terms to one side:** $$x - 2 - 8x + 48 = 0$$ $$-7x + 46 = 0$$ 5. **Solve for $x$:** $$-7x = -46$$ $$x = \frac{-46}{-7}$$ $$x = \frac{46}{7}$$ 6. **Interpretation:** The functions $f(x)$ and $g(x)$ intersect at $x = \frac{46}{7} \approx 6.57$. 7. **Verify by substitution:** $$f\left(\frac{46}{7}\right) = \frac{\frac{46}{7} - 2}{-2} = \frac{\frac{46}{7} - \frac{14}{7}}{-2} = \frac{\frac{32}{7}}{-2} = \frac{32}{7} \times \left(-\frac{1}{2}\right) = -\frac{16}{7}$$ $$g\left(\frac{46}{7}\right) = -4 \times \frac{46}{7} + 24 = -\frac{184}{7} + \frac{168}{7} = -\frac{16}{7}$$ Both values match, confirming the solution. **Final answer:** $$x = \frac{46}{7}$$