Question

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

Answer 1

Answer:

9. c. $-4.5<x<4.5$

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

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

Step-by-step 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,

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