(g) Solve for \( x \) and represent your answer on a number line: \( \begin{array}{ll}\text { (1) } 2 x+1 \geq 13 & \text { (2) } 2-2 x \leq 3 \\ \text { (3) } 3(x+7)>6 & \text { (4) } 10-2 x<22+2 x \\ \text { (5) } \frac{1-x}{2} \leq 3-x & \text { (6) } \frac{-5 x}{3}-1 \geq x+7\end{array} \)

Solve the inequality by following steps: - step0: Solve for \(x\): \(3\left(x+7\right)-6>0\) - step1: Calculate: \(3x+15>0\) - step2: Move the constant to the right side: \(3x>0-15\) - step3: Remove 0: \(3x>-15\) - step4: Divide both sides: \(\frac{3x}{3}>\frac{-15}{3}\) - step5: Divide the numbers: \(x>-5\) Solve the equation \( (-5x)/3-1-x-7\geq 0 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(\frac{\left(-5x\right)}{3}-1-x-7\geq 0\) - step1: Remove the parentheses: \(\frac{-5x}{3}-1-x-7\geq 0\) - step2: Simplify: \(-\frac{5x}{3}-8-x\geq 0\) - step3: Multiply both sides: \(\left(-\frac{5x}{3}-8-x\right)\times 3\geq 0\times 3\) - step4: Multiply the terms: \(-5x-24-3x\geq 0\) - step5: Simplify: \(-8x-24\geq 0\) - step6: Move the constant to the right side: \(-8x\geq 0+24\) - step7: Remove 0: \(-8x\geq 24\) - step8: Change the signs: \(8x\leq -24\) - step9: Divide both sides: \(\frac{8x}{8}\leq \frac{-24}{8}\) - step10: Divide the numbers: \(x\leq -3\) Solve the equation \( 10-2x-22-2x<0 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(10-2x-22-2x<0\) - step1: Subtract the terms: \(-12-4x<0\) - step2: Move the constant to the right side: \(-4x<0+12\) - step3: Remove 0: \(-4x<12\) - step4: Change the signs: \(4x>-12\) - step5: Divide both sides: \(\frac{4x}{4}>\frac{-12}{4}\) - step6: Divide the numbers: \(x>-3\) Solve the equation \( 2x+1-13\geq 0 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(2x+1-13\geq 0\) - step1: Subtract the numbers: \(2x-12\geq 0\) - step2: Move the constant to the right side: \(2x\geq 0+12\) - step3: Remove 0: \(2x\geq 12\) - step4: Divide both sides: \(\frac{2x}{2}\geq \frac{12}{2}\) - step5: Divide the numbers: \(x\geq 6\) Solve the equation \( (1-x)/2-3+x\leq 0 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(\frac{\left(1-x\right)}{2}-3+x\leq 0\) - step1: Remove the parentheses: \(\frac{1-x}{2}-3+x\leq 0\) - step2: Multiply both sides: \(\left(\frac{1-x}{2}-3+x\right)\times 2\leq 0\times 2\) - step3: Multiply the terms: \(1-x-6+2x\leq 0\) - step4: Simplify: \(-5+x\leq 0\) - step5: Move the constant to the right side: \(x\leq 0+5\) - step6: Remove 0: \(x\leq 5\) Solve the equation \( 2-2x-3\leq 0 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(2-2x-3\leq 0\) - step1: Subtract the numbers: \(-1-2x\leq 0\) - step2: Move the constant to the right side: \(-2x\leq 0+1\) - step3: Remove 0: \(-2x\leq 1\) - step4: Change the signs: \(2x\geq -1\) - step5: Divide both sides: \(\frac{2x}{2}\geq \frac{-1}{2}\) - step6: Divide the numbers: \(x\geq -\frac{1}{2}\) Let's solve each inequality step by step and summarize the results: 1. **Inequality (1):** \[ 2x + 1 \geq 13 \] Subtracting 1 from both sides: \[ 2x \geq 12 \] Dividing by 2: \[ x \geq 6 \] 2. **Inequality (2):** \[ 2 - 2x \leq 3 \] Subtracting 2 from both sides: \[ -2x \leq 1 \] Dividing by -2 (remember to flip the inequality): \[ x \geq -\frac{1}{2} \] 3. **Inequality (3):** \[ 3(x + 7) > 6 \] Expanding: \[ 3x + 21 > 6 \] Subtracting 21 from both sides: \[ 3x > -15 \] Dividing by 3: \[ x > -5 \] 4. **Inequality (4):** \[ 10 - 2x < 22 + 2x \] Rearranging: \[ 10 - 22 < 2x + 2x \] Simplifying: \[ -12 < 4x \] Dividing by 4: \[ x > -3 \] 5. **Inequality (5):** \[ \frac{1 - x}{2} \leq 3 - x \] Multiplying through by 2: \[ 1 - x \leq 6 - 2x \] Rearranging: \[ x \leq 5 \] 6. **Inequality (6):** \[ \frac{-5x}{3} - 1 \geq x + 7 \] Rearranging: \[ \frac{-5x}{3} - x \geq 8 \] Combining terms: \[ \frac{-5x - 3x}{3} \geq 8 \] Simplifying: \[ \frac{-8x}{3} \geq 8 \] Multiplying by -3 (flipping the inequality): \[ 8x \leq -24 \] Dividing by 8: \[ x \leq -3 \] ### Summary of Results: 1. \( x \geq 6 \) 2. \( x \geq -\frac{1}{2} \) 3. \( x > -5 \) 4. \( x > -3 \) 5. \( x \leq 5 \) 6. \( x \leq -3 \) ### Combined Results: - From inequalities (1) and (5): \( x \) must satisfy \( x \geq 6 \) and \( x \leq 5 \), which is not possible. - From inequalities (2) and (6): \( x \) must satisfy \( x \geq -\frac{1}{2} \) and \( x \leq -3 \), which is also not possible. ### Final Representation on a Number Line: The only valid solution is from inequality (1): - \( x \geq 6 \) This means the solution set is represented on the number line as follows: - A solid dot at 6 and a line extending to the right, indicating all values greater than or equal to 6. \[ \text{Number Line: } \text{---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---} \] \[ \text{... -5 -4 -3 -2 -1 0 1 2 3 4 5 6 \textbf{●} 7 8 9 ...} \]