Please make sure you actually do write out everything the way your instructors would like. Your instructors are responsible for making sure you get a certain level of math knowledge (assuming you do your part); that's their job. Writing out everything helps them do that. More importantly, it also helps you learn the math deeply. Of course, you are the one who benefits from all this.
This book assumes you already know algebra, and can use it to solve equations. If you are still learning algebra, that's okay - you can still benefit from this book. The only requirement is that you have some experience solving algebra equations. As you read, if there are parts that talk about math that you haven't learned yet, just skip over those sections for now.
There is a difference between mathematical knowledge, and what goes on inside of you when you apply that knowledge. This book is only about the second part. There is a bit of trickiness here. Some of the techniques in this book are, in a sense, shortcuts. You can use them to mentally bypass steps in a calculation. But if you use them to do math that you don't yet understand how to do normally, then it's possible that you will apply the techniques incorrectly. You will still get an answer, but it will probably not be mathematically correct. Even worse, you may "learn" faulty math habits.
Fortunately, there is an easy way to handle this: only use the techniques in this book to do math that you already know how to solve normally (on paper). If you are not sure, attempt to solve it on paper first. As you do this, you will get better and better at knowing when it is safe to use the techniques in this book. Following this guideline will allow you to safely get the greatest long-term value from this book.