1. The problem is to solve the system of inequalities: $$3x + 5 < 20$$ $$7x - 15 < 20$$ $$9x + 4 \geq 31$$ $$-5x + 12 > -8$$ $$-3x + 9 < 0$$ 2. We solve each inequality step-by-step using basic algebraic rules: add/subtract terms and divide by coefficients, remembering to reverse inequality signs when dividing by negative numbers. 3. Solve the first inequality: $$3x + 5 < 20$$ Subtract 5 from both sides: $$3x < 15$$ Divide both sides by 3: $$x < 5$$ 4. Solve the second inequality: $$7x - 15 < 20$$ Add 15 to both sides: $$7x < 35$$ Divide both sides by 7: $$x < 5$$ 5. Solve the third inequality: $$9x + 4 \geq 31$$ Subtract 4 from both sides: $$9x \geq 27$$ Divide both sides by 9: $$x \geq 3$$ 6. Solve the fourth inequality: $$-5x + 12 > -8$$ Subtract 12 from both sides: $$-5x > -20$$ Divide both sides by -5 (reverse inequality): $$x < 4$$ 7. Solve the fifth inequality: $$-3x + 9 < 0$$ Subtract 9 from both sides: $$-3x < -9$$ Divide both sides by -3 (reverse inequality): $$x > 3$$ 8. Combine all inequalities: From 1 and 2: $$x < 5$$ From 3: $$x \geq 3$$ From 4: $$x < 4$$ From 5: $$x > 3$$ 9. The solution must satisfy all simultaneously, so: $$3 < x < 4$$ This is the final solution set for the system of inequalities.