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Solve the inequality for [tex]$x$[/tex] and identify the graph of its solution.

[tex]|x+2| \ \textless \ 2[/tex]

Choose the answer that gives both the correct solution and the correct graph.

A. Solution: [tex]-4 \ \textless \ x \ \textless \ 0[/tex]

B. Solution: [tex]x \ \textgreater \ -4[/tex] and [tex]x \ \textless \ 0[/tex]

C. Solution: [tex]x \ \textless \ -4[/tex] or [tex]x \ \textgreater \ 0[/tex]

Sagot :

GinnyAnswer
To solve the inequality [tex]\( |x + 2| < 2 \)[/tex], we need to break it down into two separate inequalities based on the definition of absolute value. The absolute value [tex]\( |x + 2| \)[/tex] represents the distance of [tex]\( x + 2 \)[/tex] from 0, which must be less than 2. This translates into two inequalities:

1. [tex]\( x + 2 < 2 \)[/tex]
2. [tex]\( x + 2 > -2 \)[/tex]

### Solving the First Inequality:

1. [tex]\( x + 2 < 2 \)[/tex]
2. Subtract 2 from both sides:
[tex]\[ x < 0 \][/tex]

### Solving the Second Inequality:

1. [tex]\( x + 2 > -2 \)[/tex]
2. Subtract 2 from both sides:
[tex]\[ x > -4 \][/tex]

### Combining the Inequalities:

By combining the two inequalities, we obtain:
[tex]\[ -4 < x < 0 \][/tex]

This solution tells us that [tex]\( x \)[/tex] must be greater than [tex]\(-4\)[/tex] and less than [tex]\( 0 \)[/tex].

### Result and Graph:

The solution [tex]\( -4 < x < 0 \)[/tex] means that [tex]\( x \)[/tex] lies within the open interval from [tex]\(-4\)[/tex] to [tex]\( 0\)[/tex], but does not include [tex]\(-4\)[/tex] or [tex]\( 0\)[/tex] themselves.

Thus, the correct answer is:
B. Solution: [tex]\( x>-4 \)[/tex] and [tex]\( x<0 \)[/tex]
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