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Solution - Absolute value equations

Exact form:

x = \frac{3}{5}, \frac{1}{3}

x=\frac{3}{5} , \frac{1}{3}

Decimal form:

x = 0.6, 0.333

x=0.6 , 0.333

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

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

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:

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

2. Solve the two equations for x

14 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

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

Simplify the arithmetic:

$(- x + 1) = 4 x - 2$

Subtract from both sides:

$(- x + 1) - 4 x = (4 x - 2) - 4 x$

Group like terms:

$(- x - 4 x) + 1 = (4 x - 2) - 4 x$

Simplify the arithmetic:

$- 5 x + 1 = (4 x - 2) - 4 x$

Group like terms:

$- 5 x + 1 = (4 x - 4 x) - 2$

Simplify the arithmetic:

$- 5 x + 1 = - 2$

Subtract from both sides:

$(- 5 x + 1) - 1 = - 2 - 1$

Simplify the arithmetic:

$- 5 x = - 2 - 1$

Simplify the arithmetic:

$- 5 x = - 3$

Divide both sides by :

$\frac{(- 5 x)}{- 5} = \frac{- 3}{- 5}$

Cancel out the negatives:

$\frac{5 x}{5} = \frac{- 3}{- 5}$

Simplify the fraction:

$x = \frac{- 3}{- 5}$

Cancel out the negatives:

$x = \frac{3}{5}$

13 additional steps

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

Expand the parentheses:

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

Expand the parentheses:

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

Multiply the coefficients:

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

Simplify the arithmetic:

$(- x + 1) = - 4 x + 2$

Add to both sides:

$(- x + 1) + 4 x = (- 4 x + 2) + 4 x$

Group like terms:

$(- x + 4 x) + 1 = (- 4 x + 2) + 4 x$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$3 x + 1 = 2$

Subtract from both sides:

$(3 x + 1) - 1 = 2 - 1$

Simplify the arithmetic:

$3 x = 2 - 1$

Simplify the arithmetic:

$3 x = 1$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{1}{3}$

Simplify the fraction:

$x = \frac{1}{3}$

3. List the solutions

$x = \frac{3}{5}, \frac{1}{3}$
(2 solution(s))

4. Graph

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