Solve Absolute Value Equation |x^2 - 2x| = |x^2 + 6x|

In summary, to solve an absolute value equation, first divide both sides by the absolute value of the variable. Then, consider the two possible cases where the variable is positive or negative and solve for both. Finally, check the solutions to see if they satisfy the original equation. For equations with higher powers, expanding them out may be necessary but it will not eliminate the absolute value bars.
  • #1
mathdad
1,283
1
Solve the absolute value equation.

|x^2 - 2x| = |x^2 + 6x|

Seeking the first step.
 
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  • #2
$x = 0$ is one solution.
Now divide both sides by $|x|$.

Next, write the resulting equation as $|x - 2| = |x - (-6)|$.
Then you seek those $x$ which have equal distance to the points $2$ and $-6$ on the real line.
 
  • #3
|x^2 - 2x| = |x^2 + 6x|

|x(x - 2)| = |x(x + 6)|

|x||x - 2| = |x||x+6|

[|x||x - 2|]/|x| = [|x||x+6|]/|x|

|x-2| = |x+6|

x - 2 = x + 6

The only solution is x = 0.
 
  • #4
RTCNTC said:
|x^2 - 2x| = |x^2 + 6x|

|x(x - 2)| = |x(x + 6)|

|x||x - 2| = |x||x+6|

[|x||x - 2|]/|x| = [|x||x+6|]/|x|

|x-2| = |x+6|

x - 2 = x + 6

The only solution is x = 0.

You can't just remove the absolute value as if they have no meaning. If we have:

\(\displaystyle |a|=|b|\)

Then, this implies:

\(\displaystyle a=\pm b\)

And so, you have another case to evaluate:

\(\displaystyle x-2=-(x+6)\)

\(\displaystyle 2x=-4\)

\(\displaystyle x=-2\)

Let's examine the hint given by Janssens:

"Then you seek those \(x\) which have equal distance to the points 2 and −6 on the real line."

This would be the mid-point or mean of the two stated values:

\(\displaystyle x=\frac{-6+2}{2}=-2\)

And so, we conclude that the original problem has two solutions, given by:

\(\displaystyle x\in\{-2,0\}\)
 
  • #5
What if the same question involves higher powers?

Example:

|x^2 - 2x|^3 = |x^2 + 6x|^3
 
  • #6
RTCNTC said:
What if the same question involves higher powers?

Example:

|x^2 - 2x|^3 = |x^2 + 6x|^3
IF they have that as a given then you have to expand it out. But for the record \(\displaystyle |x|^3 = |x| \cdot x^2\). It won't help you to get rid of the absolute value bars.

-Dan
 
  • #7
topsquark said:
IF they have that as a given then you have to expand it out. But for the record \(\displaystyle |x|^3 = |x| \cdot x^2\). It won't help you to get rid of the absolute value bars.

-Dan
I will post a few more if needed.
 

Related to Solve Absolute Value Equation |x^2 - 2x| = |x^2 + 6x|

1. What is an absolute value equation?

An absolute value equation is an equation that contains an absolute value expression, which is represented by two vertical lines surrounding a number or variable. The absolute value of a number represents its distance from zero on a number line, so an absolute value equation is essentially asking for which values of the variable make the expression equal to a certain number.

2. How do you solve an absolute value equation?

To solve an absolute value equation, you need to isolate the absolute value expression on one side of the equation and then remove the absolute value bars. This can be done by setting up two separate equations, one with the positive value of the expression and one with the negative value, and solving for the variable in each equation. The solutions will be the values of the variable that make both equations true.

3. What are the steps to solve |x^2 - 2x| = |x^2 + 6x|?

The steps to solve this absolute value equation are:
1. Isolate the absolute value expressions on each side of the equation
2. Set up two separate equations, one with the positive value of the expression and one with the negative value
3. Solve each equation for the variable
4. Check your solutions by plugging them back into the original equation
5. Write the final solution set in interval notation.

4. Can an absolute value equation have no solutions?

Yes, an absolute value equation can have no solutions. This occurs when the absolute value expression on one side of the equation is equal to a negative number, which is impossible. In this case, the equation has no solutions and is considered to be "inconsistent".

5. Are there any special cases when solving absolute value equations?

Yes, there are two special cases when solving absolute value equations. The first is when the absolute value expressions on both sides of the equation are equal to each other, resulting in a single solution. The second is when the absolute value expression is equal to zero, resulting in two solutions (one for the positive value and one for the negative value). It is important to check for these special cases when solving absolute value equations.

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