**Problem: Solve the systems of linear inequalities given in parts (ა), (ბ), and (გ) of each group.** --- ### Group 1 **(ა)** 1. Start with inequality $3z + 2 \geq 7 + 4z$. 2. Subtract $4z$ from both sides: $3z - 4z + 2 \geq 7$. 3. Simplify: $-z + 2 \geq 7$. 4. Subtract 2 from both sides: $-z \geq 5$. 5. Multiply by $-1$ (flip inequality): $z \leq -5$. Second inequality: $4z - 1 \leq 2z + 7$. 1. Subtract $2z$ from both sides: $2z - 1 \leq 7$. 2. Add $1$ to both sides: $2z \leq 8$. 3. Divide by $2$: $z \leq 4$. **Solution:** $z \leq -5$ and $z \leq 4$ imply $z \leq -5$. **(ბ)** 1. Inequality $2x + 6 > 3x - 1$. 2. Subtract $3x$: $2x - 3x + 6 > -1$. 3. Simplify: $-x + 6 > -1$. 4. Subtract 6: $-x > -7$. 5. Multiply by $-1$ (flip inequality): $x < 7$. Second: $5x - 1 \geq 2x + 8$. 1. Subtract $2x$: $3x -1 \geq 8$. 2. Add $1$: $3x \geq 9$. 3. Divide by $3$: $x \geq 3$. **Solution:** $3 \leq x < 7$. **(გ)** 1. Inequality $2z + 8 \leq z + 4$. 2. Subtract $z$: $z + 8 \leq 4$. 3. Subtract $8$: $z \leq -4$. Second: $2z + 8 \geq z -1$. 1. Subtract $z$: $z + 8 \geq -1$. 2. Subtract $8$: $z \geq -9$. **Solution:** $-9 \leq z \leq -4$. --- ### Group 2 **(ა)** 1. $(x - 1)/2 > 1$. 2. Multiply both sides by $2$: $x - 1 > 2$. 3. Add $1$: $x > 3$. Second: $x + 3 > 0$ implies $x > -3$. **Solution:** $x > 3$ (since $x>3$ satisfies $x>-3$). **(ბ)** 1. $2x > -3$ implies $x > -3/2$. Second: $(x/8) - (x/4) \leq 1/2$. 1. Get common denominator: $x/8 - 2x/8 = -x/8 \leq 1/2$. 2. Multiply both sides by $-8$ (flip inequality): $x \geq -4$. **Solution:** $x > -1.5$ and $x \geq -4$ so solution is $x > -1.5$. **(გ)** 1. $(2x - 2)/3 \leq 1/3$. 2. Multiply both sides by $3$: $2x - 2 \leq 1$. 3. Add $2$: $2x \leq 3$. 4. Divide by $2$: $x \leq 1.5$. Second: $1 - 4x \geq 0$. 1. Subtract 1: $-4x \geq -1$. 2. Multiply by $-1$: $4x \leq 1$. 3. Divide by $4$: $x \leq 0.25$. **Solution:** $x \leq 0.25$ (since it is more restrictive than $1.5$). --- ### Group 3 System 1: 1. $5 - 2x \leq 4$. 2. Subtract 5: $-2x \leq -1$. 3. Multiply by $-1$: $2x \geq 1$. 4. Divide by $2$: $x \geq 0.5$. Next: $3x - 11.25 \leq 10.2$. 1. Add $11.25$: $3x \leq 21.45$. 2. Divide by $3$: $x \leq 7.15$. Next: $3x \geq 1 - 2x$. 1. Add $2x$: $5x \geq 1$. 2. Divide by $5$: $x \geq 0.2$. **Solution:** $x \geq 0.5$ (because $x \geq 0.5$ and $x \geq 0.2$ combine) and $x \leq 7.15$. System 2: 1. $24.5x + 20 \leq 1 - 4.5x$. 2. Add $4.5x$: $29x + 20 \leq 1$. 3. Subtract $20$: $29x \leq -19$. 4. Divide by $29$: $x \leq -19/29 \approx -0.655$. Next: $1 - x \geq 3x + 5$. 1. Subtract $3x$: $1 - 4x \geq 5$. 2. Subtract $1$: $-4x \geq 4$. 3. Multiply by $-1$: $4x \leq -4$. 4. Divide by $4$: $x \leq -1$. Next: $10.01 - 3x \leq 2 - 5x$. 1. Add $5x$: $10.01 + 2x \leq 2$. 2. Subtract $10.01$: $2x \leq -8.01$. 3. Divide by $2$: $x \leq -4.005$. **Solution:** $x \leq -4.005$ (most restrictive among $x \leq -0.655$, $x \leq -1$, and $x \leq -4.005$). System 3: 1. $5 + 2x > 9 - x$. 2. Add $x$: $5 + 3x > 9$. 3. Subtract $5$: $3x > 4$. 4. Divide by $3$: $x > 4/3 \approx 1.333$. Next: $x + 28.4 \geq 9 - 3x$. 1. Add $3x$: $4x + 28.4 \geq 9$. 2. Subtract $28.4$: $4x \geq -19.4$. 3. Divide by $4$: $x \geq -4.85$. Next: $4x - 1 > 10.4 - 2x$. 1. Add $2x$: $6x - 1 > 10.4$. 2. Add $1$: $6x > 11.4$. 3. Divide by $6$: $x > 1.9$. **Solution:** Combining $x > 1.333$ and $x > 1.9$, the solution is $x > 1.9$ (since 1.9 is stricter) and $x \geq -4.85$ (redundant). So final is $x > 1.9$. ---