The inequality to solve is:

$|2x + 7| \leq 19$

Step-by-Step Solution:

To solve the absolute value inequality $|2x + 7| \leq 19$, we split it into two inequalities:

1. $2x + 7 \leq 19$
2. $-(2x + 7) \leq 19$

Solve the first inequality:

$2x + 7 \leq 19$ Subtract 7 from both sides: $2x \leq 12$ Divide by 2: $x \leq 6$

Solve the second inequality:

$-(2x + 7) \leq 19$ Distribute the negative: $-2x - 7 \leq 19$ Add 7 to both sides: $-2x \leq 26$ Divide by -2 (flip the inequality sign): $x \geq -13$

Solution Set:

The solution is the intersection of $x \leq 6$ and $x \geq -13$, which gives the interval:

$x \in [-13, 6]$

In interval notation, the solution set is:

$[-13, 6]$

Graph:

The graph would show a closed interval from -13 to 6, with filled circles on both -13 and 6 to indicate that these values are included in the solution set.

Would you like further details or a deeper explanation?

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Here are 5 related questions:

1. How do you solve absolute value equations that involve strict inequalities like $|2x + 7| < 19$?
2. What are the steps to graph an inequality on a number line?
3. How do the rules change if the inequality sign is $\geq$ or $\leq$?
4. What happens if there is no intersection between the two parts of the inequality?
5. How can you express the union of solution sets in interval notation?

Tip: Always remember to flip the inequality sign when dividing or multiplying by a negative number in inequalities.