1. **State the problem:** We want to find which equations are equivalent to $$\frac{1}{2}(4 + 8x) = 17$$. 2. **Start with the given equation:** $$\frac{1}{2}(4 + 8x) = 17$$ 3. **Multiply both sides by 2 to eliminate the fraction:** $$\cancel{\frac{1}{2}}(4 + 8x) \times 2 = 17 \times 2$$ $$4 + 8x = 34$$ 4. **Check each option:** - Option 1: $$2 + 4x = 8.5$$ Multiply both sides by 2: $$4 + 8x = 17$$ which is not equal to 34, so this is **not equivalent**. - Option 2: $$4 + 8x = 34$$ This matches the simplified equation, so this is **equivalent**. - Option 3: $$4x = 15$$ From $$4 + 8x = 34$$, subtract 4: $$8x = 30$$ Divide both sides by 2: $$\cancel{8x \div 2} = 30 \div 2$$ $$4x = 15$$ So this is **equivalent**. - Option 4: $$2 = 17 - 8x$$ Rearranged from $$4 + 8x = 34$$: Subtract 8x from both sides: $$4 = 34 - 8x$$ Subtract 2 from both sides: $$2 = 30 - 8x$$ which is not equal to $$2 = 17 - 8x$$, so this is **not equivalent**. **Final answer:** The equations equivalent to $$\frac{1}{2}(4 + 8x) = 17$$ are: - $$4 + 8x = 34$$ - $$4x = 15$$