Answer :
We start with the inequality
[tex]$$
81 > -9f + 4.
$$[/tex]
Step 1. Subtract 4 from both sides:
Subtracting 4 from both sides, we have
[tex]$$
81 - 4 > -9f,
$$[/tex]
which simplifies to
[tex]$$
77 > -9f.
$$[/tex]
Step 2. Divide by [tex]\(-9\)[/tex]:
Since we are dividing by a negative number, we must reverse the inequality sign. Dividing both sides by [tex]\(-9\)[/tex] gives
[tex]$$
\frac{77}{-9} < f,
$$[/tex]
or equivalently,
[tex]$$
f > -\frac{77}{9}.
$$[/tex]
The fraction [tex]\(-\frac{77}{9}\)[/tex] is approximately [tex]\(-8.56\)[/tex].
Step 3. Test the candidate values:
We need to check which of the following values satisfy
[tex]$$
f > -\frac{77}{9}.
$$[/tex]
1. For [tex]$f = 10$[/tex]:
Since [tex]$10 > -8.56$[/tex], this value satisfies the inequality.
2. For [tex]$f = 7$[/tex]:
Since [tex]$7 > -8.56$[/tex], this value also satisfies the inequality.
3. For [tex]$f = -3$[/tex]:
Since [tex]$-3 > -8.56$[/tex], this value satisfies the inequality.
4. For [tex]$f = -4$[/tex]:
Since [tex]$-4 > -8.56$[/tex], this value satisfies the inequality.
Conclusion:
All the given candidate values [tex]$f=10$[/tex], [tex]$f=7$[/tex], [tex]$f=-3$[/tex], and [tex]$f=-4$[/tex] are solutions to the inequality since all of them are greater than [tex]$-\frac{77}{9}$[/tex].
Thus, the solutions are:
[tex]$$
f=10,\quad f=7,\quad f=-3,\quad f=-4.
$$[/tex]
[tex]$$
81 > -9f + 4.
$$[/tex]
Step 1. Subtract 4 from both sides:
Subtracting 4 from both sides, we have
[tex]$$
81 - 4 > -9f,
$$[/tex]
which simplifies to
[tex]$$
77 > -9f.
$$[/tex]
Step 2. Divide by [tex]\(-9\)[/tex]:
Since we are dividing by a negative number, we must reverse the inequality sign. Dividing both sides by [tex]\(-9\)[/tex] gives
[tex]$$
\frac{77}{-9} < f,
$$[/tex]
or equivalently,
[tex]$$
f > -\frac{77}{9}.
$$[/tex]
The fraction [tex]\(-\frac{77}{9}\)[/tex] is approximately [tex]\(-8.56\)[/tex].
Step 3. Test the candidate values:
We need to check which of the following values satisfy
[tex]$$
f > -\frac{77}{9}.
$$[/tex]
1. For [tex]$f = 10$[/tex]:
Since [tex]$10 > -8.56$[/tex], this value satisfies the inequality.
2. For [tex]$f = 7$[/tex]:
Since [tex]$7 > -8.56$[/tex], this value also satisfies the inequality.
3. For [tex]$f = -3$[/tex]:
Since [tex]$-3 > -8.56$[/tex], this value satisfies the inequality.
4. For [tex]$f = -4$[/tex]:
Since [tex]$-4 > -8.56$[/tex], this value satisfies the inequality.
Conclusion:
All the given candidate values [tex]$f=10$[/tex], [tex]$f=7$[/tex], [tex]$f=-3$[/tex], and [tex]$f=-4$[/tex] are solutions to the inequality since all of them are greater than [tex]$-\frac{77}{9}$[/tex].
Thus, the solutions are:
[tex]$$
f=10,\quad f=7,\quad f=-3,\quad f=-4.
$$[/tex]
