I am exploring the topic of absolute value inequalities and I am really wondering how to solve this inequality without graphing:
$$ |x|+2|x+1|-3|x+2|+4|x+3|≤7 $$
I know it is possible to solve it by considering all 16 cases for this equation, but that seems very unappealing to me. Do you guys see any smart solutions?
So, for the answer, we can reduce the cases to 4 only, we can see from inequality that, x has to be less than 0, it cannot be more than or equal to it, you can verify yourself by putting in x=0!
After a minute or two of calculation, we come to know that x cannot be less than -4, so our domain to check has reduced to: -4 < x < 0.
Within this, we can check for 4 cases:
CASE I: -1 <= x < 0 Solving in this you will get: x E [-1,-0.5] EQ1....
CASE II: -2 <= x < -1 Solving this: x E [-1.5,1) EQ2....
CASE III: -3 <= x < -2 Solving this: x E [-3,-2.25] EQ3....
CASE IV: -4 < x < -3 Solving this: x E [-3.75,-3) EQ4....
By the union of all 4 equations, we get: x E [-3.75,-2.25] U [-1.5,-0.5]