1. **State the problem:** Solve the system of inequalities: $$2 \geq 1 - 2x + 3 < 4x$$ $$-1 - 3x \leq 1 - 8x$$ $$4x \leq 2x + 8$$ 2. **Solve the first compound inequality:** Rewrite the first inequality as two parts: $$2 \geq 4 - 2x$$ $$4 - 2x < 4x$$ - For $$2 \geq 4 - 2x$$: $$2 - 4 \geq -2x$$ $$-2 \geq -2x$$ Divide both sides by $$-2$$ (remember to reverse inequality): $$1 \leq x$$ - For $$4 - 2x < 4x$$: $$4 < 6x$$ $$\frac{4}{6} < x$$ $$\frac{2}{3} < x$$ Combining both parts: $$x \geq 1$$ (since $$1 > \frac{2}{3}$$) 3. **Solve the second inequality:** $$-1 - 3x \leq 1 - 8x$$ Add $$8x$$ to both sides: $$-1 + 5x \leq 1$$ Add $$1$$ to both sides: $$5x \leq 2$$ Divide by 5: $$x \leq \frac{2}{5}$$ 4. **Solve the third inequality:** $$4x \leq 2x + 8$$ Subtract $$2x$$ from both sides: $$2x \leq 8$$ Divide by 2: $$x \leq 4$$ 5. **Combine all solutions:** From step 2: $$x \geq 1$$ From step 3: $$x \leq \frac{2}{5}$$ From step 4: $$x \leq 4$$ The second inequality requires $$x \leq \frac{2}{5}$$, but the first requires $$x \geq 1$$. Since $$1 > \frac{2}{5}$$, there is no overlap between these two inequalities. Therefore, the system has **no solution** where all inequalities hold simultaneously. **Final answer:** $$\boxed{\text{No solution}}$$