Question

Number 9, c,e, f can anyone help me solve them

Answer 1

9. c.
`-4.5<x<4.5`

9. e.
`x\in (-4,-3)\cup (-1,0)`

9. f.
`x\in (2,4)\cup (6,8)`

Explanation:

9. c. Given

`|4-|2x||<5`

Rewrite it as follows:

`||2x|-4|<5`

This inequality is equivalent ot the double inequality

`-5<|2x|-4<5`

Add 4:

`-5+4<|2x|-4+4<5+4\\ \\-1<|2x|<9`

But the absolute value |2x| is always no less than 0, so

`0\le |2x|<9\\ \\-9<2x<9\\ \\-4.5<x<4.5`

9. e. Given

`1<|x+2|<2`

This inequality is equivalent to

`\left\{\begin{array}{l}|x+2|>1\\|x+2|<2\end{array}\right.\Rightarrow \left\{\begin{array}{l}\left[\begin{array}{l}x+2>1\\x+2<-1\end{array}\right.\\-2<x+2<2\end{array}\right.\Rightarrow \left\{\begin{array}{l}\left[\begin{array}{l}x>-1\\x<-3\end{array}\right.\\-4<x<0\end{array}\right.`

So,

`x\in (-4,-3)\cup (-1,0)`

9. f. Given

`1<|2x-10|-1<5`

Add 1:

`2<|2x-10|<6`

This inequality is equivalent to

`\left\{\begin{array}{l}|2x-10|>2\\|2x-10|<6\end{array}\right.\Rightarrow \left\{\begin{array}{l}\left[\begin{array}{l}2x-10>2\\2x-10<-2\end{array}\right.\\-6<2x-10<6\end{array}\right.\Rightarrow \left\{\begin{array}{l}\left[\begin{array}{l}2x>12\\2x<8\end{array}\right.\\4<2x<16\end{array}\right.\\ \\\left\{\begin{array}{l}\left[\begin{array}{l}x>6\\x<4\end{array}\right.\\2<x<8\end{array}\right.`

So,

`x\in (2,4)\cup (6,8)`