Absolute Value Equations

Does this problem have a solution that is equal to one another?

|x + 4| = 3x + 2

Question

Absolute Value Equations 

Does this problem have a solution that is equal to one another?

|x + 4| = 3x + 2

anonymous · Answer

there is only one solution here, although sometimes there are two

anonymous · Answer

i got 2.5 = 2.5 but its supposed to be 2.5 = - 2.5 what did i do wrong?

anonymous · Answer

first of all $1$ is a solution "by instpecton" since it is pretty clear that 
$$|1+4|=3+2$$

anonymous · Answer

$2.5$ is not a solution

anonymous · Answer

through my experience with absolute value always one answer is positive and one answer is negative but they're the same number

anonymous · Answer

not true here however

anonymous · Answer

im talkin about the negative side oops sorry forgot to mention that

anonymous · Answer

?

anonymous · Answer

i have for the positive x = 1 and 5 = 5 and for the negative i have x = 1.5 and 2.5 = 2.5

anonymous · Answer

you lost me

anonymous · Answer

|x +4| = 3x + 2 is the positive and |x + 4| = -3x - 2 is the negative

anonymous · Answer

They will have the same magnitude, but opposing signs you mean.

anonymous · Answer

if $x>-4$ then $|x+4|=x+4$ and you can solve 
$$x+4=3x+2$$  easly

anonymous · Answer

you get 
$$x=1$$

anonymous · Answer

yea but the other side is negative for the second half of the problem

anonymous · Answer

and its -1.5

anonymous · Answer

careful here

anonymous · Answer

and i plug it in

anonymous · Answer

if $x<-4$ then 
$$|x+4|=-x-4$$ and lets see what happens when we solve 
$$-x-4=3x+2$$

anonymous · Answer

we get 
$$4x=-6$$ so 
$$x=-\frac{3}{2}=-1.5$$ HOWEVER

anonymous · Answer

we were making the assumption that $x<-4$ and so $x$ CANNOT  be $-1.5$ since $-1.5>-4$

anonymous · Answer

conclude that there is only one solution $x=1$
alternatively graph the line $y=3x+2$ and the V shaped $y=|x+4|$ and see that they intersect in only one place

anonymous · Answer

wait you're losing me so x cant be greater than -4?

anonymous · Answer

lets go slow

anonymous · Answer

thats what she said

anonymous · Answer

it is clear that if you solve 
$$x+4=2x+3$$ you get $x=1$ right

actually she said "if you want to make me happy you gotta make it snappy"

anonymous · Answer

yea i get that part

anonymous · Answer

https://www.youtube.com/watch?v=wiPizq1OEG8

anonymous · Answer

ok so then the other thing to solve is 
$$-x-4=3x+2$$

anonymous · Answer

but $|x+4|=-x-4$

anonymous · Answer

ONLY IF $x<-4$ 
otherwise $|x+4|=x+4$

anonymous · Answer

so when you solve 
$$-x-4=3x+2$$ and get some number LARGER than $-4$ it is not a real solution

anonymous · Answer

wait u lost me with -x-4 = 3x+2   why did you do that?

anonymous · Answer

because $|x+4|$ is either $x+4$ or its negative $-x-4$
what two equation would you solve?

anonymous · Answer

hey is that a rule? if the number is larger than the number all the way to the right that its a no solution?

anonymous · Answer

i don't know if it is a rule, but it is true

anonymous · Answer

i bet you were going to solve 
$$x+4=-3x-2$$ right?

anonymous · Answer

yea!!

anonymous · Answer

why isnt that right???

anonymous · Answer

well convince yourself that that is the same as solving 
$$-x-4=3x+2$$

anonymous · Answer

just multiply both sides by $-1$

anonymous · Answer

so go ahead, solve 
$$x+4=-3x-2$$ and when you check your answer you will see it does not work !!

anonymous · Answer

yea its not supposed to work the guy in the GRE video got 2.5 and a -2.5 thats the right answer

anonymous · Answer

im tryna figure out how he got the 2nd negative cuz i keep gettin 2.5 for both

anonymous · Answer

there is no $2.5$ in it 
i see what you mean though
once you solve, you get $$x=-1.5$$  then substitute in to the original equation

anonymous · Answer

yea!! thats what i did and it came out to be 2.5 for both sides

anonymous · Answer

but its supposed to be -2.5 on the right side .... i don't kno how he got that

anonymous · Answer

because you did not substitute in to the ORIGINAL  equation 
$$|x + 4| = 3x + 2$$ you substituted in to 
$$x+4=-3x-2$$

anonymous · Answer

it is a solution to the second equation, but not to the first

anonymous · Answer

wait i thought you were supposed to substitute the what you got from the negative to the negative problem?

anonymous · Answer

you have to plug in both x=1 and x=-1.5 to the first equation not the second?

anonymous · Answer

yes, you are solving $$|x + 4| = 3x + 2$$ right ? that is what you have to check

anonymous · Answer

forgot how much i liked george clinton

anonymous · Answer

oh wth............. i did not know that.... i see it now

anonymous · Answer

whew

anonymous · Answer

now i am going to go listen to more funk 
later

anonymous · Answer

nice !!