To solve the inequality [tex]\(-8 < 2x + 2\)[/tex], we'll test each given value one by one. Let's rewrite the inequality to make it more straightforward:
[tex]\[
-8 < 2x + 2
\][/tex]
Subtract 2 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
-8 - 2 < 2x
\][/tex]
[tex]\[
-10 < 2x
\][/tex]
Next, divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
-5 < x
\][/tex]
This means [tex]\(x\)[/tex] must be greater than [tex]\(-5\)[/tex]. Now we will test each of the given values to determine if they satisfy this inequality.
### Value I: [tex]\(x = -5\)[/tex]
Substitute [tex]\(-5\)[/tex] into the inequality:
[tex]\[
-5 < -5
\][/tex]
This statement is false because [tex]\(-5\)[/tex] is not less than [tex]\(-5\)[/tex].
### Value II: [tex]\(x = -10\)[/tex]
Substitute [tex]\(-10\)[/tex] into the inequality:
[tex]\[
-10 < -5
\][/tex]
This statement is false because [tex]\(-10\)[/tex] is not greater than [tex]\(-5\)[/tex].
### Value III: [tex]\(x = -4\)[/tex]
Substitute [tex]\(-4\)[/tex] into the inequality:
[tex]\[
-4 < -5
\][/tex]
This statement is false, but let's reconsider the manipulation context. Actually reconsider equation directly after translation:
[tex]\(-8 < 2(-4) + 2\)[/tex]
[tex]\(-8 < -8\)[/tex]
Here equality is also not fair.
Among the values given:
[tex]\(\text{III}\)[/tex] is the proper satisfactory value.
Hence, the solution to the inequality [tex]\( -8 < 2x + 2 \)[/tex] is:
\textbf{III only}
