addisonbosley2
Answered

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What is the solution to [tex]|x-5|+2\ \textless \ 20[/tex]?

A. [tex]-7 \ \textless \ x \ \textless \ 15[/tex]

B. [tex]-13 \ \textless \ x \ \textless \ 23[/tex]

C. [tex]x \ \textless \ -7[/tex] or [tex]x \ \textgreater \ 15[/tex]

D. [tex]x \ \textless \ -13[/tex] or [tex]x \ \textgreater \ 23[/tex]

Sagot :

GinnyAnswer
To solve the inequality [tex]\( |x - 5| + 2 < 20 \)[/tex], we need to first isolate the absolute value expression. Here is the detailed, step-by-step solution:

1. Start by isolating the absolute value expression on one side of the inequality:
[tex]\[ |x - 5| + 2 < 20 \][/tex]

Subtract 2 from both sides:
[tex]\[ |x - 5| < 18 \][/tex]

2. Recall that the expression [tex]\( |A| < B \)[/tex] implies [tex]\( -B < A < B \)[/tex]. Apply this property to our inequality:
[tex]\[ -18 < x - 5 < 18 \][/tex]

3. Now, solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex] in the middle part of this compound inequality. Add 5 to all three parts:
[tex]\[ -18 + 5 < x - 5 + 5 < 18 + 5 \][/tex]

Simplify the expressions:
[tex]\[ -13 < x < 23 \][/tex]

So, the solution to [tex]\( |x - 5| + 2 < 20 \)[/tex] is:
[tex]\[ -13 < x < 23 \][/tex]

Thus, the correct option is:
[tex]\[ -13 < x < 23 \][/tex]
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