First isolate the absolute value expression:
[tex]8|x-11|+3\ge57[/tex]
[tex]8|x-11|\ge54[/tex]
[tex]|x-11|\ge\dfrac{54}8=\dfrac{27}4[/tex]
Then by definition of absolute value,
- if [tex]x\ge11[/tex], then [tex]|x-11|=x-11[/tex], so the inequality reduces to [tex]x-11\ge\dfrac{27}4[/tex] and we end up with [tex]x\ge\dfrac{71}4[/tex]; otherwise,
- if [tex]x<11[/tex], then [tex]|x-11|=-(x-11)=11-x[/tex], so we have [tex]11-x\ge\dfrac{27}4[/tex], so [tex]\dfrac{17}4\ge x[/tex]
