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Solution - Absolute value equations

Exact form:

k = - 13, - \frac{11}{9}

k=-13 , -\frac{11}{9}

Mixed number form:

k = - 13, - 1 \frac{2}{9}

k=-13 , -1\frac{2}{9}

Decimal form:

k = - 13, - 1.222

k=-13 , -1.222

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

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

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:

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

2. Solve the two equations for k

7 additional steps

$(5 k + 12) = (4 k - 1)$

Subtract from both sides:

$(5 k + 12) - 4 k = (4 k - 1) - 4 k$

Group like terms:

$(5 k - 4 k) + 12 = (4 k - 1) - 4 k$

Simplify the arithmetic:

$k + 12 = (4 k - 1) - 4 k$

Group like terms:

$k + 12 = (4 k - 4 k) - 1$

Simplify the arithmetic:

$k + 12 = - 1$

Subtract from both sides:

$(k + 12) - 12 = - 1 - 12$

Simplify the arithmetic:

$k = - 1 - 12$

Simplify the arithmetic:

$k = - 13$

10 additional steps

$(5 k + 12) = - (4 k - 1)$

Expand the parentheses:

$(5 k + 12) = - 4 k + 1$

Add to both sides:

$(5 k + 12) + 4 k = (- 4 k + 1) + 4 k$

Group like terms:

$(5 k + 4 k) + 12 = (- 4 k + 1) + 4 k$

Simplify the arithmetic:

$9 k + 12 = (- 4 k + 1) + 4 k$

Group like terms:

$9 k + 12 = (- 4 k + 4 k) + 1$

Simplify the arithmetic:

$9 k + 12 = 1$

Subtract from both sides:

$(9 k + 12) - 12 = 1 - 12$

Simplify the arithmetic:

$9 k = 1 - 12$

Simplify the arithmetic:

$9 k = - 11$

Divide both sides by :

$\frac{(9 k)}{9} = \frac{- 11}{9}$

Simplify the fraction:

$k = \frac{- 11}{9}$

3. List the solutions

$k = - 13, - \frac{11}{9}$
(2 solution(s))

4. Graph

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