<s>[INST] Solving $|a+1|+|2a+5|=5$ How to solve this equation?
$$|a+1|+|2a+5|=5$$
Here is my solution but it seems that my answers are wrong.
$$|a+1|+|2a+5|=5$$
Case 1:
$$a+1+2a+5=5$$
$$a = \frac{1}{3}$$
Case 2:
$$a+1-2a-5=5$$
$$a=-11$$
 [/INST] Squaring both sides gives
$$(a+1)^2+|(a+1)(2a+5)|+(2a+5)^2=5^2,$$
or
$$|(a+1)(2a+5)|=25-(a^2+2a+1)-(4a^2+20a+25),$$
which gives
$$|(a+1)(2a+5)|=-5a^2-22a-1.$$
Squaring again, we obtain
$$(a+1)^2(2a+5)^2=(5a^2+22a+1)^2,$$
or
$$(2a^2+7a+5)^2=(5a^2+22a+1)^2,$$
or
$$(2a^2+7a+5)^2-(5a^2+22a+1)^2=0.$$
Factorising then gives
$$(-3a^2-15a+4)(7a^2+29a+6)=0,$$ from which you should be able to proceed.
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