Question 1208247: |x+4|+2|x+4|+4|x+4|+5=19
Hi tutor
I am struggling in my algebra class, particularly with absolute value. Would you please take me step by step through the process of how I would solve it?
Found 4 solutions by josgarithmetic, ikleyn, Edwin McCravy, greenestamps:
Answer by josgarithmetic(39618)   (Show Source):
Answer by ikleyn(52788)  (Show Source):
You can put this solution on YOUR website! .
|x+4| + 2|x+4| + 4|x+4| + 5 = 19.
~~~~~~~~~~~~~~~~~~~~~
It is easy.
On the left side, you have three addends with identical part |x+4|.
So, you combine like terms in the left side. Then your equation takes the form
7*|x+4| + 5 = 19.
Now move constant term 5 from left side to the right (by subtracting 5 from both sides).
You get then
7*|x+4| = 19 - 5,
or
7*|x+4| = 14.
Next, you divide both sides by 7. It gives you
|x+4| = 14/7 = 2.
Thus, the absolute value of (x+4) is 2. It means that x+4 is EITHER 2 OR -2.
Therefore, we consider two cases.
Case 1. x+4 = 2. Subtract 4 from both sides. You will get
x = 2 - 4 = -2. So, x = -2 is the solution.
Case 2. x+4 = -2. Again, subtract 4 from both sides. You will get
x = -2 - 4 = -6. So, x = -6 is another solution.
Thus, the given equation has two solutions: x= -2 and x= -6.
CHECK. First, check that that x= -2 is the solution.
You have |-2+4| = |2| = 2. ! correct !
Next, check that x= -6 is the solution.
You have |-6+4| = |-2| = 2. ! correct !
At this point, the solution is complete.
Come again to this forum soon to learn something new !
-----------------
To see many other similar introductory level absolute value equations
with detailed solutions and explanations, look into the lesson
- Absolute Value equations
in this site.
Learn the subject from there. This lesson is to start.
When you become more familiar with the subject, you may start learning
more complicated absolute value equations from other lessons in that section.
Do not hesitate to ask questions, if you need it.
To learn this subject, it is necessary to solve sufficient amount of relevant problems
on your own. It is also good to read/(to look at) solutions from other sources.
The lessons in this section do provide you this possibility /opportunity.
Answer by Edwin McCravy(20056)  (Show Source):
You can put this solution on YOUR website!
Or you can do it this way
There are two cases:
Case 1:
What's between the absolute value bars is not negative:
If what's between the absolute value bars is not negative, you
can just replace the bars with parentheses:
|x+4| + 2|x+4| + 4|x+4| + 5 = 19
(x+4) + 2(x+4) + 4(x+4) + 5 = 19
x+4 + 2x+8 + 4x+16 + 5 = 19
7x + 33 = 19
7x = -14
x = -2
Case 2:
What's between the absolute value bars is negative:
If what's between the absolute value bars is negative, you
change all the signs between the bars and replace the bars
with parentheses:
|x+4| + 2|x+4| + 4|x+4| + 5 = 19
(-x-4) + 2(-x-4) + 4(-x-4) + 5 = 19
-x - 4 - 2x - 8 - 4x - 16 + 5 = 19
-7x - 23 = 19
-7x = 42
x = -6
Answers: x = -2 or x = -6
Edwin
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
|x+4|+2|x+4|+4|x+4|+5=19
7|x+4|+5=19
7|x+4|=14
|x+4|=14/7=2
The other responses you have received use formal algebra to solve this absolute value equation to find the two solutions. Since you are struggling with this in an algebra class, that is probably the kind of solution you want.
But note that for many absolute value equations like this, another way to find the solutions is to interpret "|x-a|=b" as meaning the difference between x and a is b.
Our absolute value equation is
|x+4| = 2
or
|x-(-4)| = 2
So the two solutions are the two numbers whose difference between x and -4 is 2 -- i.e., the two numbers that are 2 units from -4 on a number line.
That's easy -- 2 units either side of -4 on a number line are -4-2 = -6 and -4+2 = -2.
ANSWERS: -6 and -2
|
|
|
