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Solve the absolute value inequality for [tex]$x$[/tex]:

[tex]|x+5| \ \textless \ 4[/tex]

Select one:
a. [tex]-9 \ \textless \ x \ \textless \ -1[/tex]
b. [tex]x \ \textless \ \frac{4}{5}[/tex]
c. [tex]x \ \textgreater \ 9[/tex] or [tex]x \ \textless \ 1[/tex]
d. [tex]x \ \textless \ -9[/tex] or [tex]x \ \textgreater \ -1[/tex]

Sagot :

GinnyAnsweridn
To solve the absolute value inequality [tex]\( |x + 5| < 4 \)[/tex], we can break it down into two separate inequalities according to the property of absolute values. Specifically, for any real number [tex]\( a \)[/tex], the inequality [tex]\( |a| < b \)[/tex] implies [tex]\( -b < a < b \)[/tex].

So, applying this to our problem:
[tex]\[ |x + 5| < 4 \][/tex]

This can be broken down into:
[tex]\[ -4 < x + 5 < 4 \][/tex]

Now, we solve the two inequalities separately.

1. Solve the inequality:
[tex]\[ -4 < x + 5 \][/tex]
Subtract 5 from both sides:
[tex]\[ -4 - 5 < x \][/tex]
[tex]\[ -9 < x \][/tex]

2. Solve the second inequality:
[tex]\[ x + 5 < 4 \][/tex]
Subtract 5 from both sides:
[tex]\[ x < 4 - 5 \][/tex]
[tex]\[ x < -1 \][/tex]

Combining these two results, we have:
[tex]\[ -9 < x < -1 \][/tex]

So, the solution to the inequality [tex]\( |x + 5| < 4 \)[/tex] is:
[tex]\[ -9 < x < -1 \][/tex]

Thus, the correct answer is:
a. [tex]\( -9 < x < -1 \)[/tex]
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