Let's solve the given inequalities step by step.
### Inequality 1: [tex]\( f + 15 > 30 \)[/tex]
1. Subtract 15 from both sides of the inequality:
[tex]\[
f + 15 - 15 > 30 - 15
\][/tex]
2. Simplify:
[tex]\[
f > 15
\][/tex]
So, the solution to the first inequality is:
[tex]\[ f > 15 \][/tex]
### Inequality 2: [tex]\( -3f - 11 > -35 \)[/tex]
1. Add 11 to both sides of the inequality:
[tex]\[
-3f - 11 + 11 > -35 + 11
\][/tex]
2. Simplify:
[tex]\[
-3f > -24
\][/tex]
3. Divide both sides by -3. Remember to reverse the inequality sign when dividing by a negative number:
[tex]\[
f < 8
\][/tex]
So, the solution to the second inequality is:
[tex]\[ f < 8 \][/tex]
### Combine Both Inequalities
We need to find the intersection of the solutions to both inequalities. The first inequality gives [tex]\( f > 15 \)[/tex], and the second inequality gives [tex]\( f < 8 \)[/tex].
Clearly, there are no values of [tex]\( f \)[/tex] that satisfy both [tex]\( f > 15 \)[/tex] and [tex]\( f < 8 \)[/tex] at the same time.
Therefore, there is no solution that satisfies both inequalities.
### Answer to the Given Ranges and Statements
1. [tex]\( 8 < f < 15 \)[/tex]: This range implies f is between 8 and 15. This range does not overlap with the solution to [tex]\( f > 15 \)[/tex] or [tex]\( f < 8 \)[/tex].
2. [tex]\( f > 15 \)[/tex]: This range is already determined by the first inequality.
3. The solution is all real numbers: This is incorrect because the inequalities don't cover all real numbers.
4. There is no solution: This is the correct conclusion as the two inequalities do not have any overlapping solutions.
### Conclusion
Based on the above analysis, there is no solution that satisfies both inequalities.
