2.2 Change the subject of \( S=\frac{1}{2} g t^{2} \) to \( t \ \). Solve for \( x \) and illustrate your answer on a number line if \( x \in \mathrm{~N}_{0} \) \( 3 x-\frac{4 x+4}{16}>\frac{3(x-1)}{4} \)

**Solution for the first part** We start with \[ S = \frac{1}{2} g t^{2}. \] To solve for \( t \), multiply both sides by \( 2 \) to isolate \( g t^2 \): \[ 2S = g t^{2}. \] Now, divide both sides by \( g \): \[ t^{2} = \frac{2S}{g}. \] Taking the square root of both sides (and noting that \( t \) is typically nonnegative) gives: \[ t = \sqrt{\frac{2S}{g}}. \] --- **Solution for the second part** We need to solve the inequality for \( x \): \[ 3x - \frac{4x + 4}{16} > \frac{3(x-1)}{4}. \] 1. **Clear the fractions:** Multiply every term by \( 16 \) (the least common denominator): \[ 16 \cdot 3x - (4x + 4) > 16 \cdot \frac{3(x-1)}{4}. \] 2. **Simplify each term:** The left-hand side becomes: \[ 48x - (4x + 4) = 48x - 4x - 4 = 44x - 4. \] The right-hand side simplifies as: \[ 16 \cdot \frac{3(x-1)}{4} = 4 \cdot 3(x-1) = 12(x-1) = 12x - 12. \] So, the inequality now is: \[ 44x - 4 > 12x - 12. \] 3. **Solve for \( x \):** Subtract \( 12x \) from both sides: \[ 44x - 12x - 4 > -12, \] which simplifies to: \[ 32x - 4 > -12. \] Add \( 4 \) to each side: \[ 32x > -8. \] Divide by \( 32 \): \[ x > -\frac{8}{32}. \] Simplify the fraction: \[ x > -\frac{1}{4}. \] 4. **Interpretation for \( x \in \mathbb{N}_0 \):** Since \( \mathbb{N}_0 \) denotes the set of nonnegative integers (\( 0, 1, 2, \ldots \)), and every nonnegative integer is greater than \(-\frac{1}{4}\), the inequality is true for: \[ x \in \{0, 1, 2, 3, \dots\}. \] 5. **Illustration on a number line:** \[ \text{Number line:} \quad 0 \quad 1 \quad 2 \quad 3 \quad \dots \] Marked as: \[ \begin{array}{cccccccccc} \dots & \quad -1 & \quad 0 & \quad 1 & \quad 2 & \quad 3 & \quad \dots \\ & \quad \text{(unshaded)} & \quad \bullet & \quad \bullet & \quad \bullet & \quad \bullet & \quad \dots \\ \end{array} \] All points starting from \( 0 \) (including \( 0 \)) are part of the solution. --- **Final Answers:** 1. For \( S = \frac{1}{2} g t^{2} \), solving for \( t \) gives: \[ t = \sqrt{\frac{2S}{g}}. \] 2. The inequality \[ 3x - \frac{4x+4}{16} > \frac{3(x-1)}{4} \] holds for all nonnegative integers: \[ x \in \{0, 1, 2, 3, \dots\}. \]