To solve the inequality [tex]\(-5 - 4(-6h + 1) > -7h - 4 - h\)[/tex], follow these steps:
### Step 1: Distribute and simplify
First, distribute the [tex]\(-4\)[/tex] on the left-hand side:
[tex]\[
-5 - 4(-6h + 1) = -5 + 24h - 4 = 24h - 9
\][/tex]
Thus, the inequality now is:
[tex]\[
24h - 9 > -7h - 4 - h
\][/tex]
### Step 2: Simplify the right-hand side
Combine like terms on the right-hand side:
[tex]\[
-7h - 4 - h = -8h - 4
\][/tex]
So now the inequality is:
[tex]\[
24h - 9 > -8h - 4
\][/tex]
### Step 3: Collect all terms involving [tex]\(h\)[/tex] on one side
Add [tex]\(8h\)[/tex] to both sides to move all [tex]\(h\)[/tex] terms to the left side:
[tex]\[
24h + 8h - 9 > -4
\][/tex]
[tex]\[
32h - 9 > -4
\][/tex]
### Step 4: Isolate the [tex]\(h\)[/tex] term
Add 9 to both sides:
[tex]\[
32h - 9 + 9 > -4 + 9
\][/tex]
[tex]\[
32h > 5
\][/tex]
### Step 5: Solve for [tex]\(h\)[/tex]
Divide both sides by 32:
[tex]\[
h > \frac{5}{32}
\][/tex]
Therefore, the solution to the inequality is:
[tex]\[
h > \frac{5}{32}
\][/tex]
In interval notation, the solution is:
[tex]\[
\left( \frac{5}{32}, \infty \right)
\][/tex]
This represents all values of [tex]\(h\)[/tex] that are greater than [tex]\(\frac{5}{32}\)[/tex].
