1. Absolute Value

$\displaystyle \mid 2x+5 \mid - \mid x+6 \mid < 2$ Find out the suitable value range for x?

2. Re: Absolute Value

Originally Posted by srirahulan
$\displaystyle \mid 2x+5 \mid - \mid x+6 \mid < 2$ Find out the suitable value range for x?
1. Use the definition of the absolute value:

$\displaystyle |2x+5| = \left \lbrace \begin{array}{ccc}2x+5, & 2x+5 \ge 0 \implies & x\ge -\frac52 \\ -(2x+5), & 2x+5 < 0 \implies & x < -\frac52 \end{array} \right.$

$\displaystyle |x+6| = \left \lbrace \begin{array}{ccc}x+6, & x+6 \ge 0 \implies & x\ge -6 \\ -(x+6), & x+6 < 0 \implies & x < -6 \end{array} \right.$

2. You'll get 3 intervals with different inequalities:

$\displaystyle (-\infty, -6)$
$\displaystyle \left[-6, -\frac52 \right)$
$\displaystyle \left[ -\frac52, +\infty \right)$

3. Solve the corresponding 3 inequalities, check the result against the domain.

4. You should come out with $\displaystyle x \in \left(- \frac{13}3 , 3 \right)$

5. Merry Christmas!

3. Re: Absolute Value

Thank You and Merry Christmas.