Exploring Absolute Value Equations: y |x| vs y |-x|
Understanding absolute value equations is a crucial aspect of algebra, often encountered in various mathematical and real-world applications. In this article, we delve into the solutions of the equation y |x| and compare it with the similar-looking equation y |-x|. Let's explore how these equations are related and their implications.
Understanding Absolute Values
In mathematics, the absolute value of a number is its distance from zero on the number line, without considering which direction from zero the number lies. The absolute value is always non-negative. For example, |5| 5 and |-5| 5. This property is key to solving absolute value equations such as |y| |x|.
Solving y |x|
When we have the equation y |x|, we need to consider two cases: x can be positive or negative. The equation can be rewritten to indicate these two possible solutions:
y x or y -x
This means that for any given value of x, y can be either the same as x or the negative of x. The absolute value function ensures that the magnitude of y is the same as the magnitude of x, but the sign can vary.
Example 1: y x or y -x
Let's consider the specific case where x 20. We can write:
y |20| which can be written as: y 20 or y -20Therefore, we have:
|20| |20| and |20| |-20|
This demonstrates how the equation y |x| can yield two possible values for y, provided x is a non-zero number.
Comparing y |x| with y |-x|
Now, let's compare this with the equation y |-x|. The absolute value of a negative number is the positive version of that number. Therefore, |-x| is always positive, just like |x|. So, the equation y |-x| can be rewritten as:
y |x|
This is identical to the equation y |x|, meaning that the solutions for y will be the same in both cases. So, for y |-20|, we have:
y 20 or y -20
Visually, this means that the graph of y |x| and y |-x| are identical, as both represent a V-shaped graph opening upwards.
Additional Example
Let's consider another example to further illustrate this concept. Suppose we have the equations:
y x - 2 y 2xTo find the solution where these two equations intersect, we can set them equal to each other:
x - 2 2x
Let's solve this equation for x:
Rearrange the equation: 2x - x -2 x -2For the second case, we set:
2x 2 - x Rearrange the equation: 2x x 2 3x 2 x 2/3So the x-values that satisfy these equations are x -2 and x 2/3. These values can then be substituted back into the original equations to find the corresponding y-values.
Conclusion
In summary, the equation y |x| and y |-x| are essentially the same, as the absolute value function treats both positive and negative x-values in the same way. The solutions to these equations are the same, and they represent a V-shaped graph opening upwards.