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Solution - Absolute value equations

Exact form:

x = \frac{6}{7}, - 6

x=\frac{6}{7} , -6

Decimal form:

x = 0.857, - 6

x=0.857 , -6

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
$3 | - x + 2 | = 4 | x |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| - x + 2 \| = 4 \| x \|$
$x = + y$	$3 (- x + 2) = 4 (x)$
$x = - y$	$3 (- x + 2) = 4 (- (x))$
$+ x = y$	$3 (- x + 2) = 4 (x)$
$- x = y$	$3 (- (- x + 2)) = 4 (x)$

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:

$\| x \| = \| y \|$	$3 \| - x + 2 \| = 4 \| x \|$
$x = + y$ , $+ x = y$	$3 (- x + 2) = 4 (x)$
$x = - y$ , $- x = y$	$3 (- x + 2) = 4 (- (x))$

2. Solve the two equations for x

14 additional steps

$3 \cdot (- x + 2) = 4 x$

Expand the parentheses:

$3 \cdot - x + 3 \cdot 2 = 4 x$

Group like terms:

$(3 \cdot - 1) x + 3 \cdot 2 = 4 x$

Multiply the coefficients:

$- 3 x + 3 \cdot 2 = 4 x$

Simplify the arithmetic:

$- 3 x + 6 = 4 x$

Subtract from both sides:

$(- 3 x + 6) - 4 x = (4 x) - 4 x$

Group like terms:

$(- 3 x - 4 x) + 6 = (4 x) - 4 x$

Simplify the arithmetic:

$- 7 x + 6 = (4 x) - 4 x$

Simplify the arithmetic:

$- 7 x + 6 = 0$

Subtract from both sides:

$(- 7 x + 6) - 6 = 0 - 6$

Simplify the arithmetic:

$- 7 x = 0 - 6$

Simplify the arithmetic:

$- 7 x = - 6$

Divide both sides by :

$\frac{(- 7 x)}{- 7} = \frac{- 6}{- 7}$

Cancel out the negatives:

$\frac{7 x}{7} = \frac{- 6}{- 7}$

Simplify the fraction:

$x = \frac{- 6}{- 7}$

Cancel out the negatives:

$x = \frac{6}{7}$

12 additional steps

$3 \cdot (- x + 2) = 4 \cdot - x$

Expand the parentheses:

$3 \cdot - x + 3 \cdot 2 = 4 \cdot - x$

Group like terms:

$(3 \cdot - 1) x + 3 \cdot 2 = 4 \cdot - x$

Multiply the coefficients:

$- 3 x + 3 \cdot 2 = 4 \cdot - x$

Simplify the arithmetic:

$- 3 x + 6 = 4 \cdot - x$

Group like terms:

$- 3 x + 6 = (4 \cdot - 1) x$

Multiply the coefficients:

$- 3 x + 6 = - 4 x$

Add to both sides:

$(- 3 x + 6) + 4 x = (- 4 x) + 4 x$

Group like terms:

$(- 3 x + 4 x) + 6 = (- 4 x) + 4 x$

Simplify the arithmetic:

$x + 6 = (- 4 x) + 4 x$

Simplify the arithmetic:

$x + 6 = 0$

Subtract from both sides:

$(x + 6) - 6 = 0 - 6$

Simplify the arithmetic:

$x = 0 - 6$

Simplify the arithmetic:

$x = - 6$

3. List the solutions

$x = \frac{6}{7}, - 6$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | - x + 2 |$
$y = 4 | x |$
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 x

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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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