Question

How to solve 8|x-11|+3 greater than or equal to 57

Answer 1

First isolate the absolute value expression:

$8|x-11|+3\ge57$

$8|x-11|\ge54$

$|x-11|\ge\dfrac{54}8=\dfrac{27}4$

Then by definition of absolute value,

• if $x\ge11$, then $|x-11|=x-11$, so the inequality reduces to $x-11\ge\dfrac{27}4$ and we end up with $x\ge\dfrac{71}4$; otherwise,
• if $x$, then $|x-11|=-(x-11)=11-x$, so we have $11-x\ge\dfrac{27}4$, so $\dfrac{17}4\ge x$