find any intercepts of the graph of the given equation. determine whether the graph of the equation possesses symmetry with respect to the x-axis, y-axis, or orgin

|x|+|y|=4

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To find the intercepts of the graph of the given equation and determine its symmetry, we can start by solving for each variable at the intercepts.

1. X-Intercepts:
The x-intercepts are the points where the graph intersects the x-axis, meaning the value of y is zero. To find the x-intercepts, we set y = 0 and solve the equation:

|x| + |0| = 4.

Since the absolute value of any number is always non-negative, we can eliminate the absolute value signs. The equation simplifies to:

|x| = 4.

Now we consider the two cases, one where x is positive and one where x is negative:

Case 1: x > 0:
In this case, the equation becomes x = 4.

Case 2: x < 0:
In this case, the equation becomes -x = 4, which can be simplified to x = -4.

Therefore, the x-intercepts are x = -4 and x = 4.

2. Y-Intercepts:
The y-intercepts are the points where the graph intersects the y-axis, meaning the value of x is zero. To find the y-intercepts, we set x = 0 and solve the equation:

|0| + |y| = 4.

Since the absolute value of any number is always non-negative, we can eliminate the absolute value signs. The equation simplifies to:

|y| = 4.

Now we consider the two cases, one where y is positive and one where y is negative:

Case 1: y > 0:
In this case, the equation becomes y = 4.

Case 2: y < 0:
In this case, the equation becomes -y = 4, which can be simplified to y = -4.

Therefore, the y-intercepts are y = -4 and y = 4.

3. Symmetry:
To determine whether the graph possesses symmetry with respect to the x-axis, y-axis, or origin, we can substitute negative values for x and y in the equation and check if the resulting equation is the same.

For symmetry with respect to the x-axis:
Substituting -x and y into the equation |x| + |y| = 4, we get:
|-x| + |y| = 4.
Which simplifies to:
|x| + |y| = 4.

Since this is the same as the original equation, the graph possesses symmetry with respect to the x-axis.

For symmetry with respect to the y-axis:
Substituting x and -y into the equation |x| + |y| = 4, we get:
|x| + |-y| = 4.
Which simplifies to:
|x| + |y| = 4.

Since this is the same as the original equation, the graph possesses symmetry with respect to the y-axis.

For symmetry with respect to the origin:
Substituting -x and -y into the equation |x| + |y| = 4, we get:
|-x| + |-y| = 4.
Which simplifies to:
|x| + |y| = 4.

Since this is different from the original equation, the graph does not possess symmetry with respect to the origin.

In summary, the intercepts of the graph are x = -4, x = 4, y = -4, and y = 4. The graph possesses symmetry with respect to the x-axis and y-axis, but not the origin.