Math Snap

STEP 1

Assumptions1. The given absolute value inequality is $|3p-4|<p+8$
. We need to solve for $p$
3. We need to graph the solution set

STEP 2

We first need to consider the two cases of the absolute value inequality. The absolute value of a number is less than another number if the original number is between the positive and negative of the other number. So, we have$$-p-8<p-4<p+8$$

STEP 3

Let's solve the left inequality first. We can start by adding $p$ to both sides to isolate the $3p$ term.
$$-8<2p-$$

STEP 4

Next, add $4$ to both sides to isolate the $2p$ term.
$$-4<2p$$

STEP 5

Finally, divide both sides by $2$ to solve for $p$.
$$-2<p$$

STEP 6

Now, let's solve the right inequality. We can start by subtracting $p$ from both sides to isolate the $3p$ term.
$$3p-4<8$$

STEP 7

Next, add $4$ to both sides to isolate the $3p$ term.
$$3p<12$$

STEP 8

Finally, divide both sides by $3$ to solve for $p$.
$$p<4$$

STEP 9

Now, we have two inequalities $-2<p$ and $p<4$. The solution to the original inequality is the intersection of these two inequalities, which is $-2<p<4$.

To graph the solution set, we draw a number line and mark the points $-2$ and $4$. Since $p$ is strictly greater than $-2$ and strictly less than $4$, we use open circles at $-2$ and $4$ to indicate that these values are not included in the solution set. Then, we shade the region between $-2$ and $4$ to represent all the values of $p$ that satisfy the inequality.
The solution set is $(-2,4)$.