We are asked to solve the absolute value inequality $|2x+1|>7$.

1. Problem Description

We are asked to solve the absolute value inequality $|2x+1|>7$.

2. Solution Steps

To solve an absolute value inequality of the form $|ax+b|>c$, where $c>0$, we can rewrite it as two separate inequalities:

$ax+b > c$ or $ax+b < -c$

In our case, $a=2$, $b=1$, and $c=7$. Therefore, we have:

$2x+1 > 7$ or $2x+1 < -7$

Now we solve each inequality separately.

For $2x+1 > 7$:

Subtract 1 from both sides:

$2x+1-1 > 7-1$

$2x > 6$

Divide both sides by 2:

$\frac{2x}{2} > \frac{6}{2}$

$x > 3$

For $2x+1 < -7$:

Subtract 1 from both sides:

$2x+1-1 < -7-1$

$2x < -8$

Divide both sides by 2:

$\frac{2x}{2} < \frac{-8}{2}$

$x < -4$

Therefore, the solution to the inequality is $x > 3$ or $x < -4$.

3. Final Answer

$x < -4$ or $x > 3$