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Solution - Absolute value equations

Exact form: i=0
i=0
See steps

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

|2i+6|+|−2i+6|=0

Add −|−2i+6| to both sides of the equation:

|2i+6|+|−2i+6|−|−2i+6|=−|−2i+6|

Simplify the arithmetic

|2i+6|=−|−2i+6|

2. Rewrite the equation without absolute value bars

Use the rules:
|x|=|y| → x=±y and |x|=|y| → ±x=y
to write all four options of the equation
|2i+6|=−|−2i+6|
without the absolute value bars:

|x|=|y||2i+6|=−|−2i+6|
x=+y(2i+6)=−(−2i+6)
x=−y(2i+6)=−−(−2i+6)
+x=y(2i+6)=−(−2i+6)
−x=y−(2i+6)=−(−2i+6)

When simplified, equations x=+y and +x=y are the same and equations x=−y and −x=y are the same, so we end up with only 2 equations:

|x|=|y||2i+6|=−|−2i+6|
x=+y , +x=y(2i+6)=−(−2i+6)
x=−y , −x=y(2i+6)=−−(−2i+6)

3. Solve the two equations for i

6 additional steps

(2i+6)=-(-2i+6)

Expand the parentheses:

(2i+6)=2i-6

Subtract from both sides:

(2i+6)-2i=(2i-6)-2i

Group like terms:

(2i-2i)+6=(2i-6)-2i

Simplify the arithmetic:

6=(2i-6)-2i

Group like terms:

6=(2i-2i)-6

Simplify the arithmetic:

6=−6

The statement is false:

6=−6

The equation is false so it has no solution.

9 additional steps

(2i+6)=-(-(-2i+6))

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

(2i+6)=-2i+6

Add to both sides:

(2i+6)+2i=(-2i+6)+2i

Group like terms:

(2i+2i)+6=(-2i+6)+2i

Simplify the arithmetic:

4i+6=(-2i+6)+2i

Group like terms:

4i+6=(-2i+2i)+6

Simplify the arithmetic:

4i+6=6

Subtract from both sides:

(4i+6)-6=6-6

Simplify the arithmetic:

4i=6−6

Simplify the arithmetic:

4i=0

Divide both sides by the coefficient:

i=0

4. Graph

Each line represents the function of one side of the equation:
y=|2i+6|
y=−|−2i+6|
The equation is true where the two lines cross.

Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

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