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What is the solution to [tex]|2x - 8| \ \textless \ 2[/tex]?

A. [tex]3 \ \textless \ x \ \textless \ 5[/tex]
B. [tex]-5 \ \textless \ x \ \textless \ -3[/tex]
C. [tex]x \ \textgreater \ 5[/tex] or [tex]x \ \textless \ 3[/tex]
D. [tex]x \ \textgreater \ -3[/tex] or [tex]x \ \textless \ -5[/tex]

Sagot :

GinnyAnsweridn
To solve the inequality [tex]\( |2x - 8| < 2 \)[/tex], let's break it down step-by-step:

1. The absolute value inequality [tex]\( |2x - 8| < 2 \)[/tex] means that the expression [tex]\( 2x - 8 \)[/tex] lies within the range of -2 and 2. This can be rewritten as two separate inequalities:

[tex]\[ -2 < 2x - 8 < 2 \][/tex]

2. We now need to solve these two inequalities separately. First, let's deal with the left inequality:

[tex]\[ -2 < 2x - 8 \][/tex]

Add 8 to both sides to isolate the term involving [tex]\( x \)[/tex]:

[tex]\[ -2 + 8 < 2x \][/tex]

[tex]\[ 6 < 2x \][/tex]

Now, divide both sides by 2 to solve for [tex]\( x \)[/tex]:

[tex]\[ 3 < x \][/tex]

So, one part of the solution is:

[tex]\[ x > 3 \][/tex]

3. Now, let's deal with the right inequality:

[tex]\[ 2x - 8 < 2 \][/tex]

Again, add 8 to both sides:

[tex]\[ 2x - 8 + 8 < 2 + 8 \][/tex]

[tex]\[ 2x < 10 \][/tex]

Divide by 2:

[tex]\[ x < 5 \][/tex]

So, the other part of the solution is:

[tex]\[ x < 5 \][/tex]

4. Combining these two parts, we get the solution to the inequality:

[tex]\[ 3 < x < 5 \][/tex]

So, the correct answer is:

A. [tex]\( 3 < x < 5 \)[/tex]
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