Let's solve each inequality step-by-step and discuss the combined solution.
### First Inequality: [tex]\( 10 + 2w \geq 22 \)[/tex]
1. Subtract 10 from both sides to isolate the term with [tex]\( w \)[/tex]:
[tex]\[
10 + 2w - 10 \geq 22 - 10
\][/tex]
[tex]\[
2w \geq 12
\][/tex]
2. Divide both sides by 2 to solve for [tex]\( w \)[/tex]:
[tex]\[
\frac{2w}{2} \geq \frac{12}{2}
\][/tex]
[tex]\[
w \geq 6
\][/tex]
### Second Inequality: [tex]\( 5w - 8 > -12 \)[/tex]
1. Add 8 to both sides to isolate the term with [tex]\( w \)[/tex]:
[tex]\[
5w - 8 + 8 > -12 + 8
\][/tex]
[tex]\[
5w > -4
\][/tex]
2. Divide both sides by 5 to solve for [tex]\( w \)[/tex]:
[tex]\[
\frac{5w}{5} > \frac{-4}{5}
\][/tex]
[tex]\[
w > -\frac{4}{5}
\][/tex]
### Combining the Solutions
The solution to the system of inequalities is the union of the two solution sets:
- From the first inequality, [tex]\( w \geq 6 \)[/tex]
- From the second inequality, [tex]\( w > -\frac{4}{5} \)[/tex]
The union of these two solutions covers all real numbers greater than [tex]\(-\frac{4}{5}\)[/tex]. In interval notation, the solution is:
[tex]\[
\left(-\frac{4}{5}, \infty\right)
\][/tex]
Therefore, the interval notation for the solution is:
[tex]\[
\left(-\frac{4}{5}, \infty\right)
\][/tex]
