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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

k = - 1 \frac{1}{2}, - \frac{3}{4}

k=-1\frac{1}{2} , -\frac{3}{4}

Decimal form:

k = - 1.5, - 0.75

k=-1.5 , -0.75

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

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

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:

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

2. Solve the two equations for k

7 additional steps

$k = (3 k + 3)$

Subtract from both sides:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 k = 3$

Divide both sides by :

$\frac{(- 2 k)}{- 2} = \frac{3}{- 2}$

Cancel out the negatives:

$\frac{2 k}{2} = \frac{3}{- 2}$

Simplify the fraction:

$k = \frac{3}{- 2}$

Move the negative sign from the denominator to the numerator:

$k = \frac{- 3}{2}$

6 additional steps

$k = - (3 k + 3)$

Expand the parentheses:

$k = - 3 k - 3$

Add to both sides:

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

Simplify the arithmetic:

$4 k = (- 3 k - 3) + 3 k$

Group like terms:

$4 k = (- 3 k + 3 k) - 3$

Simplify the arithmetic:

$4 k = - 3$

Divide both sides by :

$\frac{(4 k)}{4} = \frac{- 3}{4}$

Simplify the fraction:

$k = \frac{- 3}{4}$

3. List the solutions

$k = - \frac{3}{2}, - \frac{3}{4}$
(2 solution(s))

4. Graph

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