Question

Select all of the choices that, when put together, correctly describe the piecewisefunction for the following:

Answer 1

When we open the brackets of an absolute value we get the following:

`\begin{gathered} \lvert g(x)\rvert \\ g(x),g(x)\ge0\text{ and }-g(x),g(x)<0 \end{gathered}`

So in this case we have:

`\lvert11-3x\rvert`

After open the absolute balue we get two expressions. The first one is:

`11-3x,\text{ }11-3x>0`

And the second one is:

`-11+3x,\text{ }11-3x\leq0`

So let's work with the inequalities of each case. The one in the first case is:

`11-3x>0`

We can add 3x to both sides:

`\begin{gathered} 11-3x+3x>0+3x \\ 11>3x \end{gathered}`

And we divide both sides by 3:

`\begin{gathered} (11)/(3)>(3x)/(3) \\ (11)/(3)>x \end{gathered}`

The inequality in the second case is:

`11-3x\leq0`

We can add 3x to both sides:

`\begin{gathered} 11-3x+3x\leq0+3x \\ 11\leq3x \end{gathered}`

And divide by 3:

`\begin{gathered} (11)/(3)\leq(3x)/(3) \\ (11)/(3)\leq x \end{gathered}`

Then the two parts of the function are:

`\begin{gathered} 11-3x,x<(11)/(3) \\ -11+3x,x\ge(11)/(3) \end{gathered}`

Then the answers are the second and fourth options.