Absolute Value Equation |3x - 2|/|2x - 3| = 2

[mathdad]

Solve the absolute value equation.

|3x - 2|/|2x - 3| = 2

Solution:

|3x - 2| = 2|2x - 3|

3x - 2 = 2(2x - 3)

3x - 2 = 4x - 6

Solving for x, I get x = 4.

However, the textbook has two answers for this problem.
The answer is also 8/7.

How do I find 8/7?

 

[Olinguito]

…

Solution:

|3x - 2| = 2|2x - 3|

3x - 2 = 2(2x - 3)

…

As MarkFL pointed out in https://mathhelpboards.com/pre-algebra-algebra-2/absolute-value-equation-1-a-25152.html, you can’t just drop the absolute-value signs just like that! Otherwise you could have
$$|1|=|-1|\ \implies\ 1=-1.$$
The other solution you missed was
$$3x-2\ =\ -2(2x-3).$$
Alternatively, you can also do what you did in https://mathhelpboards.com/pre-algebra-algebra-2/absolute-value-equation-2-a-25153.html: square both sides.

 

[mathdad]

3x−2 = −2(2x−3)

3x - 2 = -4x + 6

3x + 4x = 6 + 2

7x = 8

x = 8/7

- - - Updated - - -

As MarkFL pointed out in https://mathhelpboards.com/pre-algebra-algebra-2/absolute-value-equation-1-a-25152.html, you can’t just drop the absolute-value signs just like that! Otherwise you could have
$$|1|=|-1|\ \implies\ 1=-1.$$
The other solution you missed was
$$3x-2\ =\ -2(2x-3).$$
Alternatively, you can also do what you did in https://mathhelpboards.com/pre-algebra-algebra-2/absolute-value-equation-2-a-25153.html: square both sides.

If I decide to square both sides, must I also square 2?

Like this:

|3x - 2|^2 = [2|2x - 3|]^2

or

Like this:

|3x - 2|^2 = 2[|2x - 3|]^2

 

[topsquark]

3x−2 = −2(2x−3)

3x - 2 = -4x + 6

3x + 4x = 6 + 2

7x = 8

x = 8/7

- - - Updated - - -
If I decide to square both sides, must I also square 2?

Like this:

|3x - 2|^2 = [2|2x - 3|]^2

or

Like this:

|3x - 2|^2 = 2[|2x - 3|]^2

Yes, you have to square the 2 as well. \(\displaystyle (a (x - 1))^2 = a^2 (x - 1)^2\) for example.

-Dan

 

[mathdad]

Yes, you have to square the 2 as well. \(\displaystyle (a (x - 1))^2 = a^2 (x - 1)^2\) for example.

-Dan

It really helps to know that 2 must also be squared.

 

[mathdad]

Take a look at my reply. The final quadratic equation does not factor leading to the textbook answers.

View attachment 8536

 

[MarkFL]

You didn't square the 2 on the RHS.

 

[mathdad]

You didn't square the 2 on the RHS.

You are right.

 

[mathdad]

I squared 2 on the right side but ended up with a quadratic equation that does not lead to the textbook answers. See picture.

View attachment 8537

 

[MarkFL]

You've made a sign error, you should get:

\(\displaystyle 7x^2-36x+32=0\)

 

[mathdad]

See picture for reply.

View attachment 8540