To solve the inequality [tex]\(6h - 5(h - 1) \leq 7h - 11\)[/tex], we will follow several algebraic steps. Here is the detailed, step-by-step solution:
1. Distribute the [tex]\(-5\)[/tex] within the parentheses on the left-hand side:
[tex]\[
6h - 5(h - 1) \leq 7h - 11
\][/tex]
[tex]\[
6h - 5h + 5 \leq 7h - 11
\][/tex]
2. Combine like terms on the left-hand side:
[tex]\[
(6h - 5h) + 5 \leq 7h - 11
\][/tex]
[tex]\[
h + 5 \leq 7h - 11
\][/tex]
3. Move the variable terms involving [tex]\(h\)[/tex] to one side and constants to the other side by subtracting [tex]\(7h\)[/tex] from both sides:
[tex]\[
h + 5 - 7h \leq -11
\][/tex]
[tex]\[
-6h + 5 \leq -11
\][/tex]
4. Isolate the variable term by subtracting [tex]\(5\)[/tex] from both sides:
[tex]\[
-6h + 5 - 5 \leq -11 - 5
\][/tex]
[tex]\[
-6h \leq -16
\][/tex]
5. Divide both sides of the inequality by [tex]\(-6\)[/tex] and remember to reverse the inequality sign (since dividing by a negative number reverses the inequality sign):
[tex]\[
h \geq \frac{-16}{-6}
\][/tex]
[tex]\[
h \geq \frac{16}{6}
\][/tex]
6. Simplify the fraction:
[tex]\[
h \geq \frac{16}{6} = \frac{8}{3}
\][/tex]
The solution to the inequality [tex]\(6h - 5(h - 1) \leq 7h - 11\)[/tex] is [tex]\(h \geq \frac{8}{3}\)[/tex].
In interval notation, this solution is written as:
[tex]\[
\left[\frac{8}{3}, \infty\right)
\][/tex]
Thus, the answer:
- The solution in terms of [tex]\(h\)[/tex]: [tex]\(h \geq \frac{8}{3}\)[/tex]
- The solution in interval notation: [tex]\(\left[\frac{8}{3}, \infty\right)\)[/tex]
