Question

Solve .graph the solution, state the interval notation and state the solution & set. 14x-6]

Answer 1

Starting with the inequality:

`\lvert4x-6\rvert\le4`

We have two cases:

Case 1: 4x-6≥0

Since 4x-6≥0, then |4x-6| = 4x-6. Then:

`\begin{gathered} 4x-6\le4 \\ \Rightarrow4x\le10 \\ \Rightarrow x\le(10)/(4) \\ \therefore x\le(5)/(2) \end{gathered}`

Case 2: 4x-6<0

Since 4x-6<0, then |4x-6| = -4x+6. Then:

`\begin{gathered} -4x+6\le4 \\ \Rightarrow-4x\le-2 \\ \Rightarrow x\ge(-2)/(-4) \\ \Rightarrow x\ge(1)/(2) \end{gathered}`

The solution set of the inequality is all the numbers x which are greater or equal to 1/2 AND lower or equal to 5/2:

`(1)/(2)\le x\le(5)/(2)`

The graph of the solution is a number line from 1/2 to 5/2 including the endpoints:

The interval notation of the solution, is:

`x\in\lbrack(1)/(2),(5)/(2)\rbrack`

The solution set S is:

`S=\lbrace x\in\R|(1)/(2)\le x\le(5)/(2)\rbrace`