5. Solve for \( x: \) \( \frac{3(x-2)}{2}-\frac{7(x-3)}{4} \geq 3 \) 6. Solve for \( z \) and write the answer in interval notation \( \frac{4-z}{2}-\frac{2 z-1}{3}\frac{1-x}{3}+2 \) 9. Solve for \( x \) and illustrate the answer on a number line 2x- \( \frac{x-4}{4}>\frac{3(3 x-1)}{4} \) 10. Solve for \( x \) and illustrate the answer on a number line \( -1 \leq x-3<5 \) 11. Solve for \( x \) and illustrate the answer on a number li \( -1 \leq 2 x-5 \leq 5 \)

**5. Solve for \( x \):** \[ \frac{3(x-2)}{2} - \frac{7(x-3)}{4} \geq 3 \] 1. Write both terms with a common denominator (4): \[ \frac{6(x-2)}{4} - \frac{7(x-3)}{4} \geq 3 \] 2. Combine the fractions: \[ \frac{6(x-2) - 7(x-3)}{4} \geq 3 \] 3. Expand the numerator: \[ \frac{6x - 12 - 7x + 21}{4} \geq 3 \quad \Longrightarrow \quad \frac{-x + 9}{4} \geq 3 \] 4. Multiply both sides by 4: \[ -x + 9 \geq 12 \] 5. Subtract 9 from both sides: \[ -x \geq 3 \] 6. Multiply by \(-1\) (remembering to reverse the inequality): \[ x \leq -3 \] --- **6. Solve for \( z \) and write the answer in interval notation:** \[ \frac{4-z}{2} - \frac{2z-1}{3} < z-2 \] 1. Find a common denominator (6) for the left-hand side: \[ \frac{3(4-z) - 2(2z-1)}{6} < z-2 \] 2. Distribute and simplify the numerator: \[ \frac{12 - 3z - 4z + 2}{6} = \frac{14 - 7z}{6} < z-2 \] 3. Multiply both sides by 6: \[ 14 - 7z < 6(z-2) \] 4. Expand the right-hand side: \[ 14 - 7z < 6z - 12 \] 5. Add \( 7z \) to both sides: \[ 14 < 13z - 12 \] 6. Add 12 to both sides: \[ 26 < 13z \] 7. Divide by 13: \[ 2 < z \] 8. In interval notation, the solution is: \[ (2, \infty) \] --- **7. Solve for \( x \):** \[ x - \frac{5}{2} \geq 3x + 3(x+5) \] 1. Simplify the right-hand side by distributing: \[ 3x + 3(x+5) = 3x + 3x + 15 = 6x + 15 \] 2. The inequality becomes: \[ x - \frac{5}{2} \geq 6x + 15 \] 3. Subtract \( x \) from both sides: \[ -\frac{5}{2} \geq 5x + 15 \] 4. Subtract 15 from both sides (express 15 as \(\frac{30}{2}\)): \[ -\frac{5}{2} - \frac{30}{2} \geq 5x \quad \Longrightarrow \quad -\frac{35}{2} \geq 5x \] 5. Divide both sides by 5: \[ -\frac{35}{10} \geq x \quad \Longrightarrow \quad x \leq -\frac{7}{2} \] --- **8. Solve for \( x \) and illustrate the answer on a number line:** \[ 3+\frac{x+2}{5} > \frac{1-x}{3} + 2 \] 1. Combine the constant on the right-hand side (write 2 as \(\frac{6}{3}\)): \[ \frac{1-x}{3} + 2 = \frac{1-x}{3} + \frac{6}{3} = \frac{7-x}{3} \] 2. The inequality becomes: \[ 3+\frac{x+2}{5} > \frac{7-x}{3} \] 3. Multiply every term by the least common denominator (15): \[ 15\cdot 3 + 15\cdot\frac{x+2}{5} > 15\cdot\frac{7-x}{3} \] 4. Simplify term-by-term: \[ 45 + 3(x+2) > 5(7-x) \] 5. Distribute: \[ 45 + 3x + 6 > 35 - 5x \] 6. Combine like terms: \[ 51 + 3x > 35 - 5x \] 7. Add \(5x\) to both sides: \[ 51 + 8x > 35 \] 8. Subtract 51 from both sides: \[ 8x > -16 \] 9. Divide by 8: \[ x > -2 \] *Number line illustration:* An open circle at \(-2\) with shading to the right. --- **9. Solve for \( x \) and illustrate the answer on a number line:** \[ 2x - \frac{x-4}{4} > \frac{3(3x-1)}{4} \] 1. Write \(2x\) with denominator 4: \[ 2x = \frac{8x}{4} \] So, \[ \frac{8x}{4} - \frac{x-4}{4} > \frac{9x-3}{4} \] 2. Combine the fractions on the left: \[ \frac{8x - (x-4)}{4} = \frac{8x - x + 4}{4} = \frac{7x+4}{4} > \frac{9x-3}{4} \] 3. Multiply both sides by 4: \[ 7x + 4 > 9x - 3 \] 4. Subtract \(7x\) from both sides: \[ 4 > 2x - 3 \] 5. Add 3 to both sides: \[ 7 > 2x \] 6. Divide by 2: \[ x < \frac{7}{2} \] *Number line illustration:* An open circle at \(\frac{7}{2}\) with shading to the left. --- **10. Solve for \( x \) and illustrate the answer on a number line:** \[ -1 \leq x-3 < 5 \] 1. Add 3 to all parts of the inequality: \[ -1 + 3 \leq x < 5 + 3 \] 2. Simplify: \[ 2 \leq x < 8 \] *Number line illustration:* A closed circle at 2 and an open circle at 8, shading between 2 and 8. --- **11. Solve for \( x \) and illustrate the answer on a number line:** \[ -1 \leq 2x-5 \leq 5 \] 1. Add 5 to all parts: \[ -1 + 5 \leq 2x \leq 5 + 5 \] 2. Simplify: \[ 4 \leq 2x \leq 10 \] 3. Divide all parts by 2: \[ 2 \leq x \leq 5 \] *Number line illustration:* Closed circles at 2 and 5, shading the interval between.