Question

How do you solve the graph: $1< 3x-2\le 10$?

Answer 1

We try to take points which have x coordinates that satisfies $1< 3x-2\le 10$. There is no restriction on the y coordinates. Based on the points we try to find the space or region in the 2-D plane which satisfies $1< 3x-2\le 10$. We have two inequalities in one. We break that and solve to get the required interval.

Complete step-by-step solution:
The inequation $1< 3x-2\le 10$ represents the space or region in 2-D plane where the x coordinates of points satisfy $1< 3x-2\le 10$.
Breaking the equation $1< 3x-2\le 10$, we get $1< 3x-2$ and $3x-2\le 10$.
We can solve the inequation treating them as equations for the operations like addition and subtraction. In case of multiplication and division we need to watch out for the negative values as that changes the inequality sign.
We first take some points for the x coordinates where $1<3x-2$.
We add 2 to both sides to get
$\begin{aligned} & 1+2< 3x-2+2 \\\ & \Rightarrow 3x >3 \\\ \end{aligned}$
Now we divide with positive number 3 and get
$\begin{aligned} & \dfrac{3x}{3}> \dfrac{3}{3} \\\ & \Rightarrow x >1 \\\ \end{aligned}$
The interval is $x\in \left( 1,\infty \right)$.
We now take some points for the x coordinates where $3x-2\le 10$.
We add 2 to both sides to get
$\begin{aligned} & 3x-2+2\le 10+2 \\\ & \Rightarrow 3x\le 12 \\\ \end{aligned}$
Now we divide with positive number 3 and get
$\begin{aligned} & \dfrac{3x}{3}\le \dfrac{12}{3} \\\ & \Rightarrow x\le 4 \\\ \end{aligned}$
The interval is $x\in \left( -\infty ,4 \right]$.
We need to find the combined result of $x\in \left( 1,\infty \right)$ and $x\in \left( -\infty ,4 \right]$ which gives $x\in \left( 1,4 \right]$.

Note: We can also express the inequality as the interval system where $1< 3x-2\le 10$ defines that $x\in \left( 1,4 \right]$. The interval for the y coordinates will be anything which can be defined as $y\in \left( -\infty ,\infty \right)$. We also need to remember that the points on the line $1< 3x-2\le 10$ will not be the solution for the inequation.