Let's solve each inequality step by step and determine which one has no solution.
### Inequality 1: [tex]\( 6(x+2) > x-3 \)[/tex]
1. Distribute the 6 on the left-hand side:
[tex]\[
6x + 12 > x - 3
\][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
5x + 12 > -3
\][/tex]
3. Subtract 12 from both sides:
[tex]\[
5x > -15
\][/tex]
4. Divide both sides by 5:
[tex]\[
x > -3
\][/tex]
The solution to the first inequality is [tex]\( x > -3 \)[/tex].
### Inequality 2: [tex]\( 3 + 4x \leq 2(1 + 2x) \)[/tex]
1. Distribute the 2 on the right-hand side:
[tex]\[
3 + 4x \leq 2 + 4x
\][/tex]
2. Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[
3 \leq 2
\][/tex]
This inequality simplifies to [tex]\( 3 \leq 2 \)[/tex], which is a contradiction. Therefore, there is no solution for this inequality.
### Inequality 3: [tex]\( -2(x + 6) < x - 20 \)[/tex]
1. Distribute the [tex]\(-2\)[/tex] on the left-hand side:
[tex]\[
-2x - 12 < x - 20
\][/tex]
2. Add [tex]\(2x\)[/tex] to both sides:
[tex]\[
-12 < 3x - 20
\][/tex]
3. Add 20 to both sides:
[tex]\[
8 < 3x
\][/tex]
4. Divide both sides by 3:
[tex]\[
\frac{8}{3} < x
\][/tex]
The solution to the third inequality is [tex]\( x > \frac{8}{3} \)[/tex].
### Inequality 4: [tex]\( x - 9 < 3(x - 3) \)[/tex]
1. Distribute the 3 on the right-hand side:
[tex]\[
x - 9 < 3x - 9
\][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
-9 < 2x - 9
\][/tex]
3. Add 9 to both sides:
[tex]\[
0 < 2x
\][/tex]
4. Divide both sides by 2:
[tex]\[
0 < x
\][/tex]
The solution to the fourth inequality is [tex]\( x > 0 \)[/tex].
### Conclusion
Among the four inequalities, the second inequality [tex]\( 3 + 4x \leq 2(1 + 2x) \)[/tex] has no solution, as it simplifies to a contradiction [tex]\( 3 \leq 2 \)[/tex].
