# Interesting Absolute Value problem

• flyingpig
Thank you for the very thorough explanation!In summary, the conversation is discussing the solution to the equation |5x-2| = 6x-12 and questioning why x = -14/11 is not a valid solution. By breaking the equation into two cases and considering restrictions on the values of x, it is determined that x = 10 is the only valid solution.
flyingpig

Find all x in

|5x-2| = 6x-12

## The Attempt at a Solution

http://www.wolframalpha.com/input/?i=Plot[|5x-2|+%3D+6x-12%2C{x%2C-1.5%2C1.5}]

Notice how only x = 10 is a solution?

So my question is, why isn't x = -14/11 a solution? I mean sure they don't intersect on the graph, but is there anyway of inspection on the equation to know that x = -14/11 is not a solution?

Hi, the solutions are:

5x-2=6x-12 x=10

2-5x=6x-12 x=14/11

why do you say -14/11?

flyingpig said:

Find all x in

|5x-2| = 6x-12

## The Attempt at a Solution

http://www.wolframalpha.com/input/?i=Plot[|5x-2|+%3D+6x-12%2C{x%2C-1.5%2C1.5}]

Notice how only x = 10 is a solution?

So my question is, why isn't x = -14/11 a solution? I mean sure they don't intersect on the graph, but is there anyway of inspection on the equation to know that x = -14/11 is not a solution?

Because while we are solving $5x-2=6x-12$ and $5x-2=-(6x-12)$ we also have to place restrictions of the values of x in each of these cases.

For example, in the first case we are assuming $5x-2\geq 0$ which means $x\geq 2/5$ and now if we get a value of $x<2/5$ then we have a contradiction and that solution is not valid.
For the second case where we assume $5x-2<0$ thus $x<2/5$ but we instead get an answer of $x=14/11>2/5$ this answer is invalid, so we scrap it.

Mentallic said:
Because while we are solving $5x-2=6x-12$ and $5x-2=-(6x-12)$ we also have to place restrictions of the values of x in each of these cases.

For example, in the first case we are assuming $5x-2\geq 0$ which means $x\geq 2/5$ and now if we get a value of $x<2/5$ then we have a contradiction and that solution is not valid.
For the second case where we assume $5x-2<0$ thus $x<2/5$ but we instead get an answer of $x=14/11>2/5$ this answer is invalid, so we scrap it.

But, if instead, we make fewer assumptions and take just the original formula at face value
|5x-2| = 6x-12 obviously means $0\leq 6x-12$ by the definition of absolute values. This means $x\geq 2$. This eliminates the same false answer for the OP.

Is this incorrect? And what relation do the values involving 2/5 have?

Look at this graph from http://www.wolframalpha.com/input/?i=plot+|5x-2|+,+6x-12,+x=-4..12".

A purely algebraic approach:
$$|5x-2|=\left\{\begin{array} , 5x-2&,&\text{if }\ 5x-2\ge0\\ -5x+2&,&\text{if }\ 5x-2<0\end{array}\right.\quad=\left\{\begin{array} , 5x-2&,&\text{if }\ x\ge\frac{2}{5}\\ -5x+2&,&\text{if }\ x<\frac{2}{5}\end{array}\right.$$
Case1: x ≥ 2/5, therefore |5x-2| = 5x-2

Solving |5x-2| = 6x-12 gives 5x-2 = 6x-12 which gives x = 10 .

Case2: x<2/5, therefore |5x-2| = -5x+2

Solving |5x-2| = 6x-12 gives -5x+2 = 6x-12 which gives x = 14/11 .
but 14/11 > 2/5, so it is inconsistent with this case.

Therefore, there is only one solution, x = 10 .

Last edited by a moderator:
AC130Nav said:
But, if instead, we make fewer assumptions and take just the original formula at face value
|5x-2| = 6x-12 obviously means $0\leq 6x-12$ by the definition of absolute values. This means $x\geq 2$. This eliminates the same false answer for the OP.

Is this incorrect? And what relation do the values involving 2/5 have?

No, it's correct
The method SammyS and I used by breaking it into cases and testing each one has a correlation with solving inequalities involving fractions where the unknown variable is in the denominator.

Yours is much simpler though

SammyS said:
Look at this graph from http://www.wolframalpha.com/input/?i=plot+|5x-2|+,+6x-12,+x=-4..12".

A purely algebraic approach:
$$|5x-2|=\left\{\begin{array} , 5x-2&,&\text{if }\ 5x-2\ge0\\ -5x+2&,&\text{if }\ 5x-2<0\end{array}\right.\quad=\left\{\begin{array} , 5x-2&,&\text{if }\ x\ge\frac{2}{5}\\ -5x+2&,&\text{if }\ x<\frac{2}{5}\end{array}\right.$$
Case1: x ≥ 2/5, therefore |5x-2| = 5x-2

Solving |5x-2| = 6x-12 gives 5x-2 = 6x-12 which gives x = 10 .

Case2: x<2/5, therefore |5x-2| = -5x+2

Solving |5x-2| = 6x-12 gives -5x+2 = 6x-12 which gives x = 14/11 .
but 14/11 > 2/5, so it is inconsistent with this case.

Therefore, there is only one solution, x = 10 .

You are my hero.

Last edited by a moderator:

## What is the definition of absolute value?

The absolute value of a number is its distance from zero on the number line. It is always a positive value.

## How do you solve an absolute value equation?

To solve an absolute value equation, you must isolate the absolute value expression on one side of the equation and then solve for both the positive and negative value of the expression. You can also graph the equation on a number line to find the solutions.

## What are some real-life applications of absolute value?

Absolute value is used in many real-life scenarios, such as finding the magnitude of a vector, calculating distances, and determining the error in measurements. It is also used in financial analysis to calculate the difference between a stock's current value and its previous value.

## Can absolute value be negative?

No, absolute value is always a positive value. It represents the distance from zero, which is never negative.

## How is absolute value related to inequalities?

Absolute value is often used in solving and graphing inequalities. It is used to represent the distance between two points on a number line and can help determine the solutions to the inequality.

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