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Solve [tex]|2x + 1| \leq 15[/tex].

A. [tex]x \leq 7[/tex]
B. [tex]-8 \leq x \leq 7[/tex]
C. [tex]-7 \leq x \leq 7[/tex]
D. [tex]-8 \geq x \geq 7[/tex]

Answer :

To solve the inequality [tex]\( |2x + 1| \leq 15 \)[/tex], we need to consider the definition of absolute value. The absolute value inequality [tex]\( |A| \leq B \)[/tex] means that [tex]\(-B \leq A \leq B\)[/tex]. So, let's apply this to our inequality:

1. Set up the compound inequality:

[tex]\[
-15 \leq 2x + 1 \leq 15
\][/tex]

2. Split it into two separate inequalities to solve for [tex]\( x \)[/tex]:

- Solve [tex]\( -15 \leq 2x + 1 \)[/tex]

- Solve [tex]\( 2x + 1 \leq 15 \)[/tex]

3. Solve each inequality:

- For the first inequality [tex]\( -15 \leq 2x + 1 \)[/tex]:

Subtract 1 from both sides:

[tex]\[
-15 - 1 \leq 2x
\][/tex]

[tex]\[
-16 \leq 2x
\][/tex]

Divide both sides by 2:

[tex]\[
-8 \leq x
\][/tex]

- For the second inequality [tex]\( 2x + 1 \leq 15 \)[/tex]:

Subtract 1 from both sides:

[tex]\[
2x \leq 14
\][/tex]

Divide both sides by 2:

[tex]\[
x \leq 7
\][/tex]

4. Combine the two solutions:

From the two inequalities, we get:

[tex]\[
-8 \leq x \leq 7
\][/tex]

So, the solution to the inequality [tex]\( |2x + 1| \leq 15 \)[/tex] is [tex]\(-8 \leq x \leq 7\)[/tex].

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