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Which of the following values are solutions to the inequality [tex]-8 \ \textless \ 2x + 2[/tex]?

I. -5
II. -10
III. -4

A. None
B. I only
C. II only
D. III only
E. I and II
F. I and III
G. II and III
H. I, II, and III


Sagot :

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}
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