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How do you solve for P?

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You assume the two signs the absolute can have and solve them individually: 1. 9(-(1-7p))-1=34 and 2. 9(1-7p)-1=34. Your solutions to these two individually are then the solutions for p
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Starting equation:

* 9*abs(1 - 7p) - 1 = 34

First isolate the absolute term:

* abs(1 - 7p) = (34+1)/9 = 3.889

Now you apply the inverse of absolute value, which is a function that accepts a positive value and gives you back that value and the negative version (trust me!):

* 1 - 7p = abs^-1 (3.889) = {3.889, -3.889}

Now you simply cancel the remaining terms (apply any operations on the right side to both items):

* -7p = {3.889 - 1, -3.889 - 1} = {2.889, -4.889}
* p = {2.889/(-7), -4.889/(-7)} = {-0.413, 0.698}

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Finally, we can check that our math is right by substituting back into the original:

* 9*abs(1 - 7(-0.413)) - 1 = 34
* 9*abs(1 - 7(0.698)) - 1 = 34

And you'll see that we're correct! Those two values both satisfy the equation. (Well, they would if you kept them as fractions instead of approximating them as decimals like I did, but I felt like keeping everything simple here.) Note that abs^-1 isn't actually a function since it returns two values for one input, but it is the inverse of abs, which we can see by graphing both.

I'm not sure how intuitive the step of inverting the absolute value function was, please let me know if you'd like me to explain that and make it more clear exactly what is happening!

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