Chapter ISG, Problem 5.4.2.7P

To determine

To solve the compound inequality

  

Answer to Problem 5.4.2.7P

Solution set of whole inequality: $\left\{x\right|x>6\}$

Explanation of Solution

Given information:

Inequality

  

Calculation:

  

Separating compound inequality

  $4-2x<\left|x\right|or\left|x\right|<2x-6$

First half is $4-2x<\left|x\right|$ and second half is |x|<2x-6

Solving first half,

  $4-2x<\left|x\right|$

Rewriting it,

  $\left|x\right|>4-2x$

So,

  $x<-\left(4-2x\right)orx>4-2x$

On solving,

  $x<-4+2xor x>4-2x$

Subtractingx from above compound inequality

  $(x-x)<(-4+2x-x)or(x-x)>(4-2x-x)$

On solving,

  $0<-4+x or 0>4-3x$

Rearranging

  $x>4or3x>4$

On further rearranging,

  $x>4orx>\frac{4}{3}$ Solution set of first half: $\left\{x|x>4orx>\frac{4}{3}\right\}$

Solving second half

  $\left|x\right|<2x-6.$

So,

  $-\left(2x-6\right)<x<2x-6$

On open bracket,

  $-2x+6<x<2x-6$

Separating by conjunction "and"

  $(-2x+6)<(x)and(x)<(2x-6)$

Rearranging like terms

  $-2x-x<-6and(x-2x)<-6$

On solving,

  $-3x<-6 and-x<-6$

Multiply by -1 in above compound inequality

  $-3x<-6 and-x<-6$

Dividing by 3 in 3x>6,

  $x>3andx>6$

Hence, Solution set of second half: $\left\{x\right|x>6\}$

Hence, Solution set of whole inequality: {x|x>6}

The graph of inequality is given below