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Solution - Absolute value equations

Exact form: i=-23,0
i=-\frac{2}{3} , 0
Decimal form: i=0.667,0
i=-0.667 , 0

Other Ways to Solve

Absolute value equations

Step-by-step explanation

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

|i+1|+|2i+1|=0

Add |2i+1| to both sides of the equation:

|i+1|+|2i+1||2i+1|=|2i+1|

Simplify the arithmetic

|i+1|=|2i+1|

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
|i+1|=|2i+1|
without the absolute value bars:

|x|=|y||i+1|=|2i+1|
x=+y(i+1)=(2i+1)
x=y(i+1)=(2i+1)
+x=y(i+1)=(2i+1)
x=y(i+1)=(2i+1)

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||i+1|=|2i+1|
x=+y , +x=y(i+1)=(2i+1)
x=y , x=y(i+1)=(2i+1)

3. Solve the two equations for i

10 additional steps

(i+1)=-(2i+1)

Expand the parentheses:

(i+1)=-2i-1

Add to both sides:

(i+1)+2i=(-2i-1)+2i

Group like terms:

(i+2i)+1=(-2i-1)+2i

Simplify the arithmetic:

3i+1=(-2i-1)+2i

Group like terms:

3i+1=(-2i+2i)-1

Simplify the arithmetic:

3i+1=1

Subtract from both sides:

(3i+1)-1=-1-1

Simplify the arithmetic:

3i=11

Simplify the arithmetic:

3i=2

Divide both sides by :

(3i)3=-23

Simplify the fraction:

i=-23

11 additional steps

(i+1)=-(-(2i+1))

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

(i+1)=2i+1

Subtract from both sides:

(i+1)-2i=(2i+1)-2i

Group like terms:

(i-2i)+1=(2i+1)-2i

Simplify the arithmetic:

-i+1=(2i+1)-2i

Group like terms:

-i+1=(2i-2i)+1

Simplify the arithmetic:

i+1=1

Subtract from both sides:

(-i+1)-1=1-1

Simplify the arithmetic:

i=11

Simplify the arithmetic:

i=0

Multiply both sides by :

-i·-1=0·-1

Remove the one(s):

i=0·-1

Multiply by zero:

i=0

4. List the solutions

i=-23,0
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
y=|i+1|
y=|2i+1|
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.