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Solve the following absolute value inequality:

[tex]\[ \frac{|2x + 1|}{3} \geq 5 \][/tex]

1. [tex]\( x \geq \)[/tex]
2. [tex]\( x \leq \)[/tex]

Sagot :

GinnyAnswer
To solve the absolute value inequality:
[tex]\[ \frac{|2x + 1|}{3} \geq 5 \][/tex]

Let's start by isolating the absolute value expression. Multiply both sides of the inequality by 3:
[tex]\[ |2x + 1| \geq 15 \][/tex]

The absolute value inequality [tex]\( |2x + 1| \geq 15 \)[/tex] means that the expression inside the absolute value, [tex]\( 2x + 1 \)[/tex], is either greater than or equal to 15, or less than or equal to -15.

This gives us two inequalities to solve:

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

### Solving the First Inequality
[tex]\[ 2x + 1 \geq 15 \][/tex]
Subtract 1 from both sides:
[tex]\[ 2x \geq 14 \][/tex]
Divide both sides by 2:
[tex]\[ x \geq 7 \][/tex]

### Solving the Second Inequality
[tex]\[ 2x + 1 \leq -15 \][/tex]
Subtract 1 from both sides:
[tex]\[ 2x \leq -16 \][/tex]
Divide both sides by 2:
[tex]\[ x \leq -8 \][/tex]

### Combining the Solutions
The solutions to the original inequality are the values of [tex]\( x \)[/tex] that satisfy either:
[tex]\[ x \geq 7 \quad \text{or} \quad x \leq -8 \][/tex]

Therefore, the final solution is:
[tex]\[ x \geq 7 \quad \text{or} \quad x \leq -8 \][/tex]
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