Question
Solve inequation and represent the solution set on the number line: $\frac{2 x-1}{12}-\frac{x-1}{3}<\frac{3 x+1}{4}$ where $x \in R$
✓
Answer
Given:
$\frac{2 x-1}{12}-\frac{x-1}{3}<\frac{3 x+1}{4}$, where $x \in R$.
Multiply by 12 on both sides in the above equation
$\begin{array}{l}\Rightarrow 12\left(\frac{2 x-1}{12}\right)-12\left(\frac{x-1}{3}\right)<12\left(\frac{3 x+1}{4}\right) \\ \Rightarrow(2 x-1)-4(x-1)<3(3 x+1) \\ \Rightarrow 2 x-1-4 x+4<9 x+3 \\ \Rightarrow 3-2 x<9 x+3\end{array}$
Now, subtracting 3 on both sides in the above equation
$\begin{array}{l}\Rightarrow 3-2 x-3<9 x+3-3 \\ \Rightarrow-2 x<9 x\end{array}$
Now, subtracting 9x from both the sides in the above equation
$\begin{array}{l}\Rightarrow-2 x-9 x<9 x-9 x \\ \Rightarrow-11 x<0\end{array}$
Multiplying -1 on both the sides in above equation
$\begin{array}{l}\Rightarrow(-11 x)(-1)>(0)(-1) \\ \Rightarrow 11 x>0\end{array}$
Dividing both sides by 11 in above equation
$\Rightarrow \frac{11 x}{11}>\frac{0}{11}$
Therefore,
$\Rightarrow>x>0$
$\frac{2 x-1}{12}-\frac{x-1}{3}<\frac{3 x+1}{4}$, where $x \in R$.
Multiply by 12 on both sides in the above equation
$\begin{array}{l}\Rightarrow 12\left(\frac{2 x-1}{12}\right)-12\left(\frac{x-1}{3}\right)<12\left(\frac{3 x+1}{4}\right) \\ \Rightarrow(2 x-1)-4(x-1)<3(3 x+1) \\ \Rightarrow 2 x-1-4 x+4<9 x+3 \\ \Rightarrow 3-2 x<9 x+3\end{array}$
Now, subtracting 3 on both sides in the above equation
$\begin{array}{l}\Rightarrow 3-2 x-3<9 x+3-3 \\ \Rightarrow-2 x<9 x\end{array}$
Now, subtracting 9x from both the sides in the above equation
$\begin{array}{l}\Rightarrow-2 x-9 x<9 x-9 x \\ \Rightarrow-11 x<0\end{array}$
Multiplying -1 on both the sides in above equation
$\begin{array}{l}\Rightarrow(-11 x)(-1)>(0)(-1) \\ \Rightarrow 11 x>0\end{array}$
Dividing both sides by 11 in above equation
$\Rightarrow \frac{11 x}{11}>\frac{0}{11}$
Therefore,
$\Rightarrow>x>0$
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