Question

I have a practice problem that I need answered, thank you

Answer 1

Given inequality:

`6x^2-x\text{ }<\text{ 2}`

Re-arranging:

`6x^2-x\text{ - 2 }<\text{ 0}`

Factorizing the expression to the left:

`\begin{gathered} 6x^2-2x\text{ + x -2 }<\text{ 0} \\ 6x^2\text{ -4x +3x -2 }<\text{ 0} \\ (3x-2)(2x+1)\text{ }<\text{ 0} \end{gathered}`

Hence:

`\begin{gathered} 3x-2\text{ }<\text{ 0} \\ 3x\text{ }<\text{ 2} \\ x\text{ }<\text{ }(2)/(3) \end{gathered}`

Since their product is negative. one of the factors would be positive.

`\begin{gathered} 2x\text{ + 1 > 0} \\ 2x\text{ > -1} \\ (2x)/(2)\text{ >}-\text{ }(1)/(2) \\ x\text{ > -}(1)/(2) \end{gathered}`

The solution on a number line:

The solution on interval notation:

`\mleft(-(1)/(2),\: (2)/(3)\mright)`