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Solution - Absolute value equations

Exact form:

k = 0, 0

k=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
$2 | k | = | 4 k |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| k \| = \| 4 k \|$
$x = + y$	$2 (k) = (4 k)$
$x = - y$	$2 (k) = - (4 k)$
$+ x = y$	$2 (k) = (4 k)$
$- x = y$	$2 (- (k)) = (4 k)$

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:

$\| x \| = \| y \|$	$2 \| k \| = \| 4 k \|$
$x = + y$ , $+ x = y$	$2 (k) = (4 k)$
$x = - y$ , $- x = y$	$2 (k) = - (4 k)$

2. Solve the two equations for k

3 additional steps

$2 k = 4 k$

Subtract from both sides:

$(2 k) - 4 k = (4 k) - 4 k$

Simplify the arithmetic:

$- 2 k = (4 k) - 4 k$

Simplify the arithmetic:

$- 2 k = 0$

Divide both sides by the coefficient:

$k = 0$

6 additional steps

$2 k = - 4 k$

Divide both sides by :

$\frac{(2 k)}{2} = \frac{(- 4 k)}{2}$

Simplify the fraction:

$k = \frac{(- 4 k)}{2}$

Simplify the fraction:

$k = - 2 k$

Add to both sides:

$k + 2 k = (- 2 k) + 2 k$

Simplify the arithmetic:

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

Simplify the arithmetic:

$3 k = 0$

Divide both sides by the coefficient:

$k = 0$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = 2 | k |$
$y = | 4 k |$
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 k

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 k

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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