To solve the system of inequalities, we need to tackle each inequality step by step.
### Step-by-Step Solution
Inequality 1: [tex]\(7h - 8(h - 1) \leq 3h - 12\)[/tex]
1. Distribute the [tex]\(-8\)[/tex]:
[tex]\[
7h - 8h + 8 \leq 3h - 12
\][/tex]
2. Combine like terms:
[tex]\[
-h + 8 \leq 3h - 12
\][/tex]
3. Move the terms involving [tex]\( h \)[/tex] to one side of the inequality:
[tex]\[
-h - 3h + 8 \leq -12
\][/tex]
[tex]\[
-4h + 8 \leq -12
\][/tex]
4. Isolate the term involving [tex]\( h \)[/tex]:
[tex]\[
-4h \leq -20
\][/tex]
5. Divide both sides by [tex]\(-4\)[/tex] (Note that the direction of the inequality will reverse when dividing by a negative number):
[tex]\[
h \geq 5
\][/tex]
This shows that [tex]\( h \)[/tex] must be at least 5 to satisfy the first inequality.
Inequality 2: [tex]\(h \geq 5\)[/tex]
This inequality is already in a form that's easy to interpret: [tex]\( h \)[/tex] must be at least 5.
### Combining the Inequalities
Both inequalities give us the same condition: [tex]\( h \geq 5 \)[/tex]. Therefore, the solution to the system of inequalities is [tex]\( h \geq 5 \)[/tex].
### Interval Notation
Expressing [tex]\( h \geq 5 \)[/tex] in interval notation:
[tex]\[
[5, \infty)
\][/tex]
### Graph on the Number Line
To represent the solution graphically on the number line:
1. Draw a number line.
2. Locate the point [tex]\(5\)[/tex] on the number line.
3. Draw a closed circle or bracket at [tex]\(5\)[/tex] (indicating that [tex]\( 5 \)[/tex] is included in the solution).
4. Shade the line to the right of [tex]\( 5 \)[/tex], extending towards positive infinity.
Here’s a representation:
[tex]\[
\begin{array}{ccccccccccccccccccccccccc}
\bullet & \text{------} \\
5
\end{array}
\][/tex]
Therefore, the solution in interval notation is:
[tex]\[
[5, \infty)
\][/tex]
