Please verify my approach to this question

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CHiMER4

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Solve for x using the definition of the absolute value if:
5x-|2x-3|=8

My approach:

|2x-3|={2x-3 if x>0 or -(2x-3) if x<0

-|2x-3|=8-5x
Therefore, |2x-3|=5x-8

2x-3={5x-8 if x≥3/2 or -(5x-8) if x<3/2 //not sure if I wrote this correctly

Therefore 2x-3=5x-8
x=5/3 (valid)
Or

2x-3=-(5x-8)
x=11/7 (invalid)
 
CHiMER4 said:
Therefore 2x-3=5x-8
x=5/3 (valid)
Or

2x-3=-(5x-8)
x=11/7 (invalid)
Click to expand...
Why valid? Why invalid?

I would do it like this:
Given |2x-3|=5x-8 we have 2 cases:
a) 2x-3 >= 0. Therefore 2x-3=5x-8. Solve this system (inequality and equation) for x.
b) 2x-3 < 0. Therefore -(2x-3) = 5x-8. Solve this system for x.
 
[math]\text {If } 2x - 3 < 0 \text {, then } x < \text {WHAT, and } |2x - 3| = \text {WHAT}[/math]
You are not asked about the value |x|. You do care about the value of x that is relevant to the absolute value of
2x - 3, but you do not care about the absolute value of x itself.

You say if x > 0 that |2x - 3| = 2x - 3. But 1 > 0 and 2 * 1 - 3 = - 1. The absolute value of - 1 is 1, which is not equal to
2 * 1 - 3.

Do you see where you went off the rails?
 
CHiMER4 said:
Solve for x using the definition of the absolute value if:
5x-|2x-3|=8

My approach:

|2x-3|={2x-3 if x>0 or -(2x-3) if x<0

-|2x-3|=8-5x
Therefore, |2x-3|=5x-8

2x-3={5x-8 if x≥3/2 or -(5x-8) if x<3/2 //not sure if I wrote this correctly

Therefore 2x-3=5x-8
x=5/3 (valid)
Or

2x-3=-(5x-8)
x=11/7 (invalid)
Click to expand...
Your work is correct, though I would write it a little differently to make it clear (much like lev888's).

The solution x = 5/3 is "valid" in the sense that it is within the interval where the equation you are solving in that part of the work applies; the solution x = 11/7 is "invalid" because it does not. So the only actual solution is x = 5/3.

This agrees with this graph of the LHS of the equation, which crosses y = 8 only at x = 5/3; the "invalid solution" 11/7 is where the other part of the graph, if extended, would cross y = 8.

1629043706056.png
 
lev888 said:
Why valid? Why invalid?

I would do it like this:
Given |2x-3|=5x-8 we have 2 cases:
a) 2x-3 >= 0. Therefore 2x-3=5x-8. Solve this system (inequality and equation) for x.
b) 2x-3 < 0. Therefore -(2x-3) = 5x-8. Solve this system for x.
Click to expand...
Okay, thanks
 
Y
JeffM said:
[math]\text {If } 2x - 3 < 0 \text {, then } x < \text {WHAT, and } |2x - 3| = \text {WHAT}[/math]
You are not asked about the value |x|. You do care about the value of x that is relevant to the absolute value of
2x - 3, but you do not care about the absolute value of x itself.

You say if x > 0 that |2x - 3| = 2x - 3. But 1 > 0 and 2 * 1 - 3 = - 1. The absolute value of - 1 is 1, which is not equal to
2 * 1 - 3.

Do you see where you went off the rails?
Click to expand...
Yes I do see it now, thank you
 
For future readers, a re-statement of lev's approach, when solving an equation of the form

| f | = c

where symbols f and c are expressions representing Real numbers and c is a positive number (or zero, in a trivial case).

You can remove the absolute-value signs and write

f = c [imath]\quad[/imath] or [imath]\quad[/imath] f = -c

In other words, consider both cases for f:
c, and the opposite of c

In this thread, we could write:

|2x - 3| = 5x - 8

2x - 3 = 5x - 8 [imath]\quad[/imath] or [imath]\quad[/imath] 2x - 3 = 8 - 5x

Solve each equation, to get candidates x = 5/3 or x = 11/7

As always, check your candidates! We find 11/7 to be an extraneous solution and toss it.

?
 
Last edited:
why is 11/7 invalid it is a valid solution to the equation is there any other piece of information that make it extraneous.
 
AbdelRahmanShady said:
why is 11/7 invalid it is a valid solution to the equation is there any other piece of information that make it extraneous.
Click to expand...
Did you actually check it in the equation?

5x - |2x - 3| = 8​
​
5(11/7) - |2(11/7) - 3| = 8 ?​
​
55/7 - |22/7 - 21/7| = 8 ?​
​
55/7 - |1/7| = 8 ?

55/7 - 1/7 = 8 ?

54/7 = 7 5/7, not 8​

It would be a solution to 5x + |2x - 3| = 8.

Why do you think it's valid?
 
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