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Solution - Absolute value equations

Exact form:

k = \frac{14}{3}, - \frac{10}{13}

k=\frac{14}{3} , -\frac{10}{13}

Mixed number form:

k = 4 \frac{2}{3}, - \frac{10}{13}

k=4\frac{2}{3} , -\frac{10}{13}

Decimal form:

k = 4.667, - 0.769

k=4.667 , -0.769

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
$| 5 k + 12 | = 2 | 4 k - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 k + 12 \| = 2 \| 4 k - 1 \|$
$x = + y$	$(5 k + 12) = 2 (4 k - 1)$
$x = - y$	$(5 k + 12) = 2 (- (4 k - 1))$
$+ x = y$	$(5 k + 12) = 2 (4 k - 1)$
$- x = y$	$- (5 k + 12) = 2 (4 k - 1)$

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:

$\| x \| = \| y \|$	$\| 5 k + 12 \| = 2 \| 4 k - 1 \|$
$x = + y$ , $+ x = y$	$(5 k + 12) = 2 (4 k - 1)$
$x = - y$ , $- x = y$	$(5 k + 12) = 2 (- (4 k - 1))$

2. Solve the two equations for k

14 additional steps

$(5 k + 12) = 2 \cdot (4 k - 1)$

Expand the parentheses:

$(5 k + 12) = 2 \cdot 4 k + 2 \cdot - 1$

Multiply the coefficients:

$(5 k + 12) = 8 k + 2 \cdot - 1$

Simplify the arithmetic:

$(5 k + 12) = 8 k - 2$

Subtract from both sides:

$(5 k + 12) - 8 k = (8 k - 2) - 8 k$

Group like terms:

$(5 k - 8 k) + 12 = (8 k - 2) - 8 k$

Simplify the arithmetic:

$- 3 k + 12 = (8 k - 2) - 8 k$

Group like terms:

$- 3 k + 12 = (8 k - 8 k) - 2$

Simplify the arithmetic:

$- 3 k + 12 = - 2$

Subtract from both sides:

$(- 3 k + 12) - 12 = - 2 - 12$

Simplify the arithmetic:

$- 3 k = - 2 - 12$

Simplify the arithmetic:

$- 3 k = - 14$

Divide both sides by :

$\frac{(- 3 k)}{- 3} = \frac{- 14}{- 3}$

Cancel out the negatives:

$\frac{3 k}{3} = \frac{- 14}{- 3}$

Simplify the fraction:

$k = \frac{- 14}{- 3}$

Cancel out the negatives:

$k = \frac{14}{3}$

13 additional steps

$(5 k + 12) = 2 \cdot (- (4 k - 1))$

Expand the parentheses:

$(5 k + 12) = 2 \cdot (- 4 k + 1)$

Expand the parentheses:

$(5 k + 12) = 2 \cdot - 4 k + 2 \cdot 1$

Multiply the coefficients:

$(5 k + 12) = - 8 k + 2 \cdot 1$

Simplify the arithmetic:

$(5 k + 12) = - 8 k + 2$

Add to both sides:

$(5 k + 12) + 8 k = (- 8 k + 2) + 8 k$

Group like terms:

$(5 k + 8 k) + 12 = (- 8 k + 2) + 8 k$

Simplify the arithmetic:

$13 k + 12 = (- 8 k + 2) + 8 k$

Group like terms:

$13 k + 12 = (- 8 k + 8 k) + 2$

Simplify the arithmetic:

$13 k + 12 = 2$

Subtract from both sides:

$(13 k + 12) - 12 = 2 - 12$

Simplify the arithmetic:

$13 k = 2 - 12$

Simplify the arithmetic:

$13 k = - 10$

Divide both sides by :

$\frac{(13 k)}{13} = \frac{- 10}{13}$

Simplify the fraction:

$k = \frac{- 10}{13}$

3. List the solutions

$k = \frac{14}{3}, - \frac{10}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 k + 12 |$
$y = 2 | 4 k - 1 |$
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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