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Solution - Absolute value equations

Exact form:

x = - 1, - \frac{1}{4}

x=-1 , -\frac{1}{4}

Decimal form:

x = - 1, - 0.25

x=-1 , -0.25

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
$2 | 2 x - 1 | = 3 | 4 x + 2 |$
without the absolute value bars:

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

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:

$\| x \| = \| y \|$	$2 \| 2 x - 1 \| = 3 \| 4 x + 2 \|$
$x = + y$ , $+ x = y$	$2 (2 x - 1) = 3 (4 x + 2)$
$x = - y$ , $- x = y$	$2 (2 x - 1) = 3 (- (4 x + 2))$

2. Solve the two equations for x

18 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

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

Simplify the arithmetic:

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

Expand the parentheses:

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

Multiply the coefficients:

$4 x - 2 = 12 x + 3 \cdot 2$

Simplify the arithmetic:

$4 x - 2 = 12 x + 6$

Subtract from both sides:

$(4 x - 2) - 12 x = (12 x + 6) - 12 x$

Group like terms:

$(4 x - 12 x) - 2 = (12 x + 6) - 12 x$

Simplify the arithmetic:

$- 8 x - 2 = (12 x + 6) - 12 x$

Group like terms:

$- 8 x - 2 = (12 x - 12 x) + 6$

Simplify the arithmetic:

$- 8 x - 2 = 6$

Add to both sides:

$(- 8 x - 2) + 2 = 6 + 2$

Simplify the arithmetic:

$- 8 x = 6 + 2$

Simplify the arithmetic:

$- 8 x = 8$

Divide both sides by :

$\frac{(- 8 x)}{- 8} = \frac{8}{- 8}$

Cancel out the negatives:

$\frac{8 x}{8} = \frac{8}{- 8}$

Simplify the fraction:

$x = \frac{8}{- 8}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 8}{8}$

Simplify the fraction:

$x = - 1$

18 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

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

Simplify the arithmetic:

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

Expand the parentheses:

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

Expand the parentheses:

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

Multiply the coefficients:

$4 x - 2 = - 12 x + 3 \cdot - 2$

Simplify the arithmetic:

$4 x - 2 = - 12 x - 6$

Add to both sides:

$(4 x - 2) + 12 x = (- 12 x - 6) + 12 x$

Group like terms:

$(4 x + 12 x) - 2 = (- 12 x - 6) + 12 x$

Simplify the arithmetic:

$16 x - 2 = (- 12 x - 6) + 12 x$

Group like terms:

$16 x - 2 = (- 12 x + 12 x) - 6$

Simplify the arithmetic:

$16 x - 2 = - 6$

Add to both sides:

$(16 x - 2) + 2 = - 6 + 2$

Simplify the arithmetic:

$16 x = - 6 + 2$

Simplify the arithmetic:

$16 x = - 4$

Divide both sides by :

$\frac{(16 x)}{16} = \frac{- 4}{16}$

Simplify the fraction:

$x = \frac{- 4}{16}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 4)}{(4 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{4}$

3. List the solutions

$x = - 1, - \frac{1}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 2 | 2 x - 1 |$
$y = 3 | 4 x + 2 |$
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 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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