Solving absolute value equations

In summary, when solving absolute value equations, it is important to consider all valid intervals and evaluate each absolute value separately. In some cases, the equation may reduce to one with no solution, while in others, there may be a valid solution. It is not enough to just look at the valid intervals, as the equation may still have no solution within those intervals.
  • #1
autodidude
333
0
Is this a correct a way of thinking for solving absolute value equations? Say I have |2x+6|-|x+3|=|x| and want to solve for x, then I have:

For |2x+6|
2x + 6 if x ≥ -3
-2x - 6 if x < -3

For |x+3|
x+3 if x ≥ -3
-x-3 if x<-3

For |x|
x if x ≥ 0
-x if x < 0

Am I supposed to look at the cases where x is in a valid interval? e.g. I can't have 2x+6-(x+3) = x because x can't be equal to or greater than both 0 and 3.

If this is the case, then why can't I have (-2x-6) - (-x+3) = -x? This is where x<-3 and x<0, isn't this valid? If x is less than -3 then it's also less than 0
 
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  • #2
Yes, that is also a valid equation for x. You just have to look at each case, evaluating one absolute value at a time.
If |2x+6|-|x+3|=|x|, then 2x + 6 - |x + 3| = |x| when x >= -3 and -2x - 6 - |x + 3| = |x| when x < -3.
Now we evaluate |x + 3| in this pair of equations to get four possibilities:
2x + 6 - x - 3 = |x| when x >=-3 and x >= - 3
2x + 6 + x + 3 = |x| when x >=-3 and x < - 3
-2x - 6 - x - 3 = |x| when x < -3 and x >= - 3
-2x - 6 + x + 3 = |x| when x < -3 and x < -3
Out of the four, only the first and last equations correspond to real values of x. Now we evaluate the |x| in each of those two equations to get four possibilities:
2x + 6 - x - 3 = x when x >= -3 and x >= 0
2x + 6 - x - 3 = -x when x >= -3 and x < 0
-2x - 6 + x + 3 = x when x < -3 and x >= 0
-2x - 6 + x + 3 = -x when x < -3 and x < 0
The third equation does not correspond to any values of x, so we now have 3 equations without absolute values whose solutions are the same as those of the original equation with absolute values.
x + 3 = x when x >= 0
x + 3 = -x when x >= -3 and x < 0
-x - 3 = -x when x < -3
Since the first and last equation have no solutions, the middle equation contains the only valid solution for the original equation.
 
  • #3
The left side can be simplified. |2x+6| - |x+3| = |x+3|.

So you have |x+3| = |x|.

You have 3 cases.
x < -3, -x -3 = -x, (no solution)
-3 < x < 0, x + 3 = -x or x = -3/2
0 < x, x+3 = x, (no solution)
 
  • #4
Thank you for the replies, I don't want to sound arrogant or anything but at the moment I'm kind of more interested in whether the way I currently think of it is accepted and if it is, why the last part of my initial post doesn't make sense (aside from the fact that you just can't get a solution when you solve it, I'm more interested in the intervals)
 
  • #5
autodidude said:
Is this a correct a way of thinking for solving absolute value equations? Say I have |2x+6|-|x+3|=|x| and want to solve for x, then I have:

For |2x+6|
2x + 6 if x ≥ -3
-2x - 6 if x < -3

For |x+3|
x+3 if x ≥ -3
-x-3 if x<-3

For |x|
x if x ≥ 0
-x if x < 0
Yes, that's all true.

Am I supposed to look at the cases where x is in a valid interval? e.g. I can't have 2x+6-(x+3) = x because x can't be equal to or greater than both 0 and 3.

If this is the case, then why can't I have (-2x-6) - (-x+3) = -x? This is where x<-3 and x<0, isn't this valid? If x is less than -3 then it's also less than 0
That's perfectly valid. If x< -3, then it is also less than 0 so all three of |2x+6|= -2x- 6, |x+3|= -x-3, and |x|= -x. The equation becomes -2x- 6- (-x- 3)= -x. That gives -x- 9= -x, which, since the two "-x" terms cancel, reduces to -9= 0 which is false for all x. Therefore, there is NO x<-3 satisfying the equation.

If -3< x< 0, |2x+ 6|= 2x+ 6 and |x+3|= x+ 3 but |x| is still -x. The equation becomes 2x+ 6- (x+3)= -x. That gives x+ 3= -x which reduces to 2x= -3= or x= -3/2. Since that is between -3 and 0, that is a valid solution: |2x+6|-|x+3|= |-3+ 6|-|-3/2|= 3- 3/2= 3/2= |-3/2|.

If 0< x, |2x+ 6|= 2x+ 6, |x+3|= x+ 3, and |x|= x. The equation becomes 2x+ 6- (x+3)= x. That gives x+ 3= x or 3= 0. Again, that is not true so there is no x larger than 0 that satisfies the equation.

It is not a matter of "x< -3" not being valid- it is simply that, in that case, the equation reduces to one that has no solution.
 
  • #6
^ Ah ok, I was thinking all valid intervals yielded valid equations...thank you
 

What is an absolute value equation?

An absolute value equation is an equation that contains an absolute value expression. An absolute value expression is a mathematical expression that represents the distance of a number from zero on a number line. It is denoted by |x|, where x is the number.

How do you solve absolute value equations?

To solve an absolute value equation, you need to isolate the absolute value expression on one side of the equation and then use the definition of absolute value to remove the absolute value bars. This will result in two possible solutions, one positive and one negative. You will need to check both solutions in the original equation to determine which one is the correct solution.

What is the difference between an absolute value equation and an absolute value inequality?

An absolute value equation is an equation that equates an absolute value expression to a specific value, whereas an absolute value inequality is an inequality that compares an absolute value expression to a specific value using symbols such as < or >. The solutions to an absolute value equation are finite, while the solutions to an absolute value inequality are infinite.

How do you graph absolute value equations?

To graph an absolute value equation, you can first create a table of values by choosing different values for x and calculating the corresponding values for the absolute value expression. Then plot these points on a coordinate plane and connect them to create a V-shaped graph. The vertex of the V-shaped graph is the point where the absolute value expression equals zero.

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

Absolute value equations can be used to solve problems related to distance, such as finding the shortest distance between two points or calculating the displacement of an object. They can also be used in financial planning, for example, to calculate the minimum amount of money needed to reach a certain savings goal. Additionally, absolute value equations are used in physics, engineering, and other scientific fields to model and analyze various phenomena.

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