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Solution - Absolute value equations

Exact form:

k = \frac{16}{11}, 0

k=\frac{16}{11} , 0

Mixed number form:

k = 1 \frac{5}{11}, 0

k=1\frac{5}{11} , 0

Decimal form:

k = 1.455, 0

k=1.455 , 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
$2 | 6 k - 4 | = | k + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| 6 k - 4 \| = \| k + 8 \|$
$x = + y$	$2 (6 k - 4) = (k + 8)$
$x = - y$	$2 (6 k - 4) = - (k + 8)$
$+ x = y$	$2 (6 k - 4) = (k + 8)$
$- x = y$	$2 (- (6 k - 4)) = (k + 8)$

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:

$\| x \| = \| y \|$	$2 \| 6 k - 4 \| = \| k + 8 \|$
$x = + y$ , $+ x = y$	$2 (6 k - 4) = (k + 8)$
$x = - y$ , $- x = y$	$2 (6 k - 4) = - (k + 8)$

2. Solve the two equations for k

12 additional steps

$2 \cdot (6 k - 4) = (k + 8)$

Expand the parentheses:

$2 \cdot 6 k + 2 \cdot - 4 = (k + 8)$

Multiply the coefficients:

$12 k + 2 \cdot - 4 = (k + 8)$

Simplify the arithmetic:

$12 k - 8 = (k + 8)$

Subtract from both sides:

$(12 k - 8) - k = (k + 8) - k$

Group like terms:

$(12 k - k) - 8 = (k + 8) - k$

Simplify the arithmetic:

$11 k - 8 = (k + 8) - k$

Group like terms:

$11 k - 8 = (k - k) + 8$

Simplify the arithmetic:

$11 k - 8 = 8$

Add to both sides:

$(11 k - 8) + 8 = 8 + 8$

Simplify the arithmetic:

$11 k = 8 + 8$

Simplify the arithmetic:

$11 k = 16$

Divide both sides by :

$\frac{(11 k)}{11} = \frac{16}{11}$

Simplify the fraction:

$k = \frac{16}{11}$

12 additional steps

$2 \cdot (6 k - 4) = - (k + 8)$

Expand the parentheses:

$2 \cdot 6 k + 2 \cdot - 4 = - (k + 8)$

Multiply the coefficients:

$12 k + 2 \cdot - 4 = - (k + 8)$

Simplify the arithmetic:

$12 k - 8 = - (k + 8)$

Expand the parentheses:

$12 k - 8 = - k - 8$

Add to both sides:

$(12 k - 8) + k = (- k - 8) + k$

Group like terms:

$(12 k + k) - 8 = (- k - 8) + k$

Simplify the arithmetic:

$13 k - 8 = (- k - 8) + k$

Group like terms:

$13 k - 8 = (- k + k) - 8$

Simplify the arithmetic:

$13 k - 8 = - 8$

Add to both sides:

$(13 k - 8) + 8 = - 8 + 8$

Simplify the arithmetic:

$13 k = - 8 + 8$

Simplify the arithmetic:

$13 k = 0$

Divide both sides by the coefficient:

$k = 0$

3. List the solutions

$k = \frac{16}{11}, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 2 | 6 k - 4 |$
$y = | k + 8 |$
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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