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Solution - Absolute value equations

Exact form:

x = \frac{2}{3}, 0

x=\frac{2}{3} , 0

Decimal form:

x = 0.667, 0

x=0.667 , 0

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

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

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:

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

2. Solve the two equations for x

11 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 3 x + 1 = - 1$

Subtract from both sides:

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

Simplify the arithmetic:

$- 3 x = - 1 - 1$

Simplify the arithmetic:

$- 3 x = - 2$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

$x = \frac{2}{3}$

8 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$x + 1 = 1$

Subtract from both sides:

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

Simplify the arithmetic:

$x = 1 - 1$

Simplify the arithmetic:

$x = 0$

3. List the solutions

$x = \frac{2}{3}, 0$
(2 solution(s))

4. Graph

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