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The solution is \( x \geq 6 \).
Solución
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.
\[
\text{Number Line: } \text{---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---}
\]
\[
\text{... -5 -4 -3 -2 -1 0 1 2 3 4 5 6 \textbf{●} 7 8 9 ...}
\]
Respondido por UpStudy AI y revisado por un tutor profesional

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