Answer:
In interval notation, the solution of |3 x-2|<7 is
or
.
Step-by-step explanation:
The given absolute value function is |3 x-2|<7.
It is required to solve the inequality and express the solution using interval notation. -B<x-A<B and solving them separately for x
Step 1 of 3
Given absolute value equation is |3 x-2|<7.
It can be written as -7<3 x-2<7.
To solve for the equality, 3x-2=7 and
![$$3 x-2=-7$$](https://tex.z-dn.net/?f=%24%243%20x-2%3D-7%24%24)
First, solve the equation 3x-2=7, then add 2 on both sides.
![$$\begin{aligned}&3 x=7+2 \\&3 x=9\end{aligned}$$](https://tex.z-dn.net/?f=%24%24%5Cbegin%7Baligned%7D%263%20x%3D7%2B2%20%5C%5C%263%20x%3D9%5Cend%7Baligned%7D%24%24)
Step 2 of 3
Simplify 3x=9 further, by dividing each side with 3 .
![$$\begin{aligned}&\frac{3 x}{3}=\frac{9}{3} \\&x=3\end{aligned}$$](https://tex.z-dn.net/?f=%24%24%5Cbegin%7Baligned%7D%26%5Cfrac%7B3%20x%7D%7B3%7D%3D%5Cfrac%7B9%7D%7B3%7D%20%5C%5C%26x%3D3%5Cend%7Baligned%7D%24%24)
Step 3 of 3
Similarly, 3x-2=-7
From the above term 3x-2=-7,
Add 2 on each side.
![$$\begin{aligned}&3 x=-7+2 \\&3 x=-5\end{aligned}$$](https://tex.z-dn.net/?f=%24%24%5Cbegin%7Baligned%7D%263%20x%3D-7%2B2%20%5C%5C%263%20x%3D-5%5Cend%7Baligned%7D%24%24)
Simplify $3 x=-5$ further, by dividing each side with 3 .
![$$\begin{aligned}&\frac{3 x}{3}=-\frac{5}{3} \\&x=-\frac{5}{3}\end{aligned}$$](https://tex.z-dn.net/?f=%24%24%5Cbegin%7Baligned%7D%26%5Cfrac%7B3%20x%7D%7B3%7D%3D-%5Cfrac%7B5%7D%7B3%7D%20%5C%5C%26x%3D-%5Cfrac%7B5%7D%7B3%7D%5Cend%7Baligned%7D%24%24)
Therefore, the solution is
or ![$\left\{-\frac{5}{3}, 3\right\}$](https://tex.z-dn.net/?f=%24%5Cleft%5C%7B-%5Cfrac%7B5%7D%7B3%7D%2C%203%5Cright%5C%7D%24)