To determine which inequality has no solution, we will analyze each inequality step-by-step.
### Inequality 1: [tex]\(6(x + 2) > x - 3\)[/tex]
1. Distribute the 6 on the left 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 by 5:
[tex]\[
x > -3
\][/tex]
This inequality has a solution [tex]\(x > -3\)[/tex].
### Inequality 2: [tex]\(3 + 4x \leq 2(1 + 2x)\)[/tex]
1. Distribute the 2 on the right side:
[tex]\[
3 + 4x \leq 2 + 4x
\][/tex]
2. Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[
3 \leq 2
\][/tex]
This is a contradiction because 3 is never less than or equal to 2. Therefore, this inequality has no solution.
### Inequality 3: [tex]\(-2(x + 6) < x - 20\)[/tex]
1. Distribute the [tex]\(-2\)[/tex] on the left 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 by 3:
[tex]\[
\frac{8}{3} < x
\][/tex]
or
[tex]\[
x > \frac{8}{3}
\][/tex]
This inequality has a solution [tex]\(x > \frac{8}{3}\)[/tex].
### Inequality 4: [tex]\(x - 9 < 3(x - 3)\)[/tex]
1. Distribute the 3 on the right 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 by 2:
[tex]\[
0 < x
\][/tex]
or
[tex]\[
x > 0
\][/tex]
This inequality has a solution [tex]\(x > 0\)[/tex].
### Conclusion
The inequality that has no solution is:
[tex]\[ 3 + 4x \leq 2(1 + 2x) \][/tex]
