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Solution - Absolute value equations

Exact form:

x = 0, 0

x=0 , 0

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| x | = 2 | x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x \| = 2 \| x \|$
$x = + y$	$(x) = 2 (x)$
$x = - y$	$(x) = 2 (- (x))$
$+ x = y$	$(x) = 2 (x)$
$- x = y$	$- (x) = 2 (x)$

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:

$\| x \| = \| y \|$	$\| x \| = 2 \| x \|$
$x = + y$ , $+ x = y$	$(x) = 2 (x)$
$x = - y$ , $- x = y$	$(x) = 2 (- (x))$

2. Solve the two equations for x

5 additional steps

$x = 2 x$

Subtract from both sides:

$x - 2 x = (2 x) - 2 x$

Simplify the arithmetic:

$- x = (2 x) - 2 x$

Simplify the arithmetic:

$- x = 0$

Multiply both sides by :

$- x \cdot - 1 = 0 \cdot - 1$

Remove the one(s):

$x = 0 \cdot - 1$

Multiply by zero:

$x = 0$

5 additional steps

$x = 2 \cdot - x$

Group like terms:

$x = (2 \cdot - 1) x$

Multiply the coefficients:

$x = - 2 x$

Add to both sides:

$x + 2 x = (- 2 x) + 2 x$

Simplify the arithmetic:

$3 x = (- 2 x) + 2 x$

Simplify the arithmetic:

$3 x = 0$

Divide both sides by the coefficient:

$x = 0$

3. List the solutions

$x = 0, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x |$
$y = 2 | x |$
The equation is true where the two lines cross.

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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.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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