Solve and graph the solution set on a number line.

|5x + 7| - 2 < 4

<------------------------>
-5 -4 -3 -2 -1 0 1 2 3 4

To solve the inequality |5x + 7| - 2 < 4, we will first isolate the absolute value expression and then solve it step by step.

Step 1: Remove the -2 from both sides of the inequality:
|5x + 7| - 2 + 2 < 4 + 2
|5x + 7| < 6

Step 2: Split into two cases, one for positive and one for negative:
Case 1: 5x + 7 is positive:
5x + 7 < 6

Case 2: 5x + 7 is negative:
-(5x + 7) < 6

Step 3: Solve each case separately:

Case 1: 5x + 7 < 6
Subtract 7 from both sides:
5x < 6 - 7
5x < -1
Divide both sides by 5 (since 5 is positive, the direction of the inequality does not change):
x < -1/5

Case 2: -(5x + 7) < 6
Multiply both sides by -1 (since multiplying by a negative number flips the inequality sign):
5x + 7 > -6
Subtract 7 from both sides:
5x > -6 - 7
5x > -13
Divide both sides by 5 (since 5 is positive, the direction of the inequality does not change):
x > -13/5

Step 4: Combine the solutions by taking their intersection:
The solution set is x such that x < -1/5 or x > -13/5.

To graph this solution set on a number line, draw an open circle at -1/5 (since it's less than) and shade the line to the left of -1/5 (since x < -1/5).
Then draw an open circle at -13/5 (since it's greater than) and shade the line to the right of -13/5 (since x > -13/5).

The final number line should have an open circle at -1/5 on the left and an open circle at -13/5 on the right, with shading in between and outside of the circles.

To solve the inequality |5x + 7| - 2 < 4, we can break it down into two separate cases: when 5x + 7 is positive and when it is negative.

Case 1: 5x + 7 is positive:
In this case, the absolute value function simplifies to 5x + 7 - 2 < 4.
By solving this inequality, we get 5x + 5 < 4.
Subtracting 5 from both sides gives us 5x < -1.
Dividing both sides by 5, we find x < -1/5.

Case 2: 5x + 7 is negative:
In this case, the absolute value function simplifies to -(5x + 7) - 2 < 4.
Solving this inequality, we have -5x - 7 - 2 < 4.
By combining like terms and simplifying, we get -5x - 9 < 4.
Adding 9 to both sides gives us -5x < 13.
Dividing both sides by -5 and reversing the inequality sign (since we are dividing by a negative number), we find x > -13/5.

Therefore, the solution to the inequality |5x + 7| - 2 < 4 is x < -1/5 or x > -13/5.

Now, let's graph the solution set on a number line:

<----------------------O-----O----------------------------------->
-3 -2 -1 0 1 2 3 4 5

The solution set is represented by the intervals (-∞, -13/5) U (-1/5, ∞).

|5x + 7| - 2 < 4

|5x + 7| < 6
5x+7 < 6 AND -5x - 7 < 6
5x < -1 AND -5x < 13
x < -1/5 AND x > -13/5

-13/5 < x < -1/5

draw "open" dots at -13/5 and -1/5 and join with a solid line