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Solution - Absolute value equations

Exact form:

k = 1, - \frac{5}{13}

k=1 , -\frac{5}{13}

Decimal form:

k = 1, - 0.385

k=1 , -0.385

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
$| 6 k + 3 | = | 7 k + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 k + 3 \| = \| 7 k + 2 \|$
$x = + y$	$(6 k + 3) = (7 k + 2)$
$x = - y$	$(6 k + 3) = - (7 k + 2)$
$+ x = y$	$(6 k + 3) = (7 k + 2)$
$- x = y$	$- (6 k + 3) = (7 k + 2)$

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:

$\| x \| = \| y \|$	$\| 6 k + 3 \| = \| 7 k + 2 \|$
$x = + y$ , $+ x = y$	$(6 k + 3) = (7 k + 2)$
$x = - y$ , $- x = y$	$(6 k + 3) = - (7 k + 2)$

2. Solve the two equations for k

10 additional steps

$(6 k + 3) = (7 k + 2)$

Subtract from both sides:

$(6 k + 3) - 7 k = (7 k + 2) - 7 k$

Group like terms:

$(6 k - 7 k) + 3 = (7 k + 2) - 7 k$

Simplify the arithmetic:

$- k + 3 = (7 k + 2) - 7 k$

Group like terms:

$- k + 3 = (7 k - 7 k) + 2$

Simplify the arithmetic:

$- k + 3 = 2$

Subtract from both sides:

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

Simplify the arithmetic:

$- k = 2 - 3$

Simplify the arithmetic:

$- k = - 1$

Multiply both sides by :

$- k \cdot - 1 = - 1 \cdot - 1$

Remove the one(s):

$k = - 1 \cdot - 1$

Simplify the arithmetic:

$k = 1$

10 additional steps

$(6 k + 3) = - (7 k + 2)$

Expand the parentheses:

$(6 k + 3) = - 7 k - 2$

Add to both sides:

$(6 k + 3) + 7 k = (- 7 k - 2) + 7 k$

Group like terms:

$(6 k + 7 k) + 3 = (- 7 k - 2) + 7 k$

Simplify the arithmetic:

$13 k + 3 = (- 7 k - 2) + 7 k$

Group like terms:

$13 k + 3 = (- 7 k + 7 k) - 2$

Simplify the arithmetic:

$13 k + 3 = - 2$

Subtract from both sides:

$(13 k + 3) - 3 = - 2 - 3$

Simplify the arithmetic:

$13 k = - 2 - 3$

Simplify the arithmetic:

$13 k = - 5$

Divide both sides by :

$\frac{(13 k)}{13} = \frac{- 5}{13}$

Simplify the fraction:

$k = \frac{- 5}{13}$

3. List the solutions

$k = 1, - \frac{5}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 k + 3 |$
$y = | 7 k + 2 |$
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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