- #1
mathdad
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Solve the absolute value equation.
|x^2 - 2x| = |x^2 + 6x|
Seeking the first step.
|x^2 - 2x| = |x^2 + 6x|
Seeking the first step.
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.
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.RTCNTC said:What if the same question involves higher powers?
Example:
|x^2 - 2x|^3 = |x^2 + 6x|^3
I will post a few more if needed.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
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.
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.
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.
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".
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.