Problem 1
The first step is to have just an absolute value is less than a number.
Add 5 to both sides.
3|p| < 12
We still have 3 times the absolute value, so now we divide both sides by 3.
|p| < 4
Now we have just the absolute value of an expression is less than a non-negative number.
Now we change this into a compound inequality. A compound inequality is two inequalities separated by the word "and or "or" depending on the case.
In this case, when you have the absolute value of an expression is less than a number, you separate into the expression is greater than the opposite of the number and the expression is less than the number.
In general, let X be an expression in x, and k is a non-negative number. You have |X| < k. You separate it like this:
X > -k and X < k
Now look at your problem. You have |p| < 4. The expression in absolute value is just p, so we get:
p > - 4 and p < 4
This can also be written as
-4 < p < 4,
or, p is between -4 and 4.
Now look for a graph that represents p > - 4 and p < 4. Since you have just "less than" and "greater than" but not "less than or equal to" and "greater than or equal to", you use open dots on -4 and 4. You need all numbers that are greater than -4 and less than 4 without including -4 and 4. You need open dots on -4 and 4, and the space in between -4 and 4.
The answer is the third choice.
Problem 2
This is the case in which you end up with a compound inequality using the word "or."
If X is an expression in x, and k is a non-negative number, and you have
|X| > k,
you separate it into two inequalities linked by the word "or" this way:
X < -k or X > k
Let's look at your problem.
|6p + 3| > 15
You have an expression in variable p inside the absolute value. That absolute value expression is grater than the non-negative number 15.
You separate inequalities like this:
6p + 3 < -15 or 6p + 3 > 15
Now solve each inequality always keeping the word "or" in between them.
6p < - 18 or 6p > 12
p < - 3 or p > 2
The answer is the third choice.
