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Solution - Absolute value equations

Exact form:

x = - 1, 1.645

x=-1 , 1.645

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
$| - x + 3.1 | = | - 2.1 x + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - x + 3.1 \| = \| - 2.1 x + 2 \|$
$x = + y$	$(- x + 3.1) = (- 2.1 x + 2)$
$x = - y$	$(- x + 3.1) = - (- 2.1 x + 2)$
$+ x = y$	$(- x + 3.1) = (- 2.1 x + 2)$
$- x = y$	$- (- x + 3.1) = (- 2.1 x + 2)$

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:

$\| x \| = \| y \|$	$\| - x + 3.1 \| = \| - 2.1 x + 2 \|$
$x = + y$ , $+ x = y$	$(- x + 3.1) = (- 2.1 x + 2)$
$x = - y$ , $- x = y$	$(- x + 3.1) = - (- 2.1 x + 2)$

2. Solve the two equations for x

10 additional steps

$(- x + 3.1) = (- 2.1 x + 2)$

Add to both sides:

$(- x + 3.1) + 2.1 x = (- 2.1 x + 2) + 2.1 x$

Group like terms:

$(- x + 2.1 x) + 3.1 = (- 2.1 x + 2) + 2.1 x$

Simplify the arithmetic:

$1.1 x + 3.1 = (- 2.1 x + 2) + 2.1 x$

Group like terms:

$1.1 x + 3.1 = (- 2.1 x + 2.1 x) + 2$

Simplify the arithmetic:

$1.1 x + 3.1 = 2$

Subtract from both sides:

$(1.1 x + 3.1) - 3.1 = 2 - 3.1$

Simplify the arithmetic:

$1.1 x = 2 - 3.1$

Simplify the arithmetic:

$1.1 x = - 1.1$

Divide both sides by :

$\frac{(1.1 x)}{1.1} = \frac{- 1.1}{1.1}$

Simplify the arithmetic:

$x = \frac{- 1.1}{1.1}$

Simplify the arithmetic:

$x = - 1$

13 additional steps

$(- x + 3.1) = - (- 2.1 x + 2)$

Expand the parentheses:

$(- x + 3.1) = 2.1 x - 2$

Subtract from both sides:

$(- x + 3.1) - 2.1 x = (2.1 x - 2) - 2.1 x$

Group like terms:

$(- x - 2.1 x) + 3.1 = (2.1 x - 2) - 2.1 x$

Simplify the arithmetic:

$- 3.1 x + 3.1 = (2.1 x - 2) - 2.1 x$

Group like terms:

$- 3.1 x + 3.1 = (2.1 x - 2.1 x) - 2$

Simplify the arithmetic:

$- 3.1 x + 3.1 = - 2$

Subtract from both sides:

$(- 3.1 x + 3.1) - 3.1 = - 2 - 3.1$

Simplify the arithmetic:

$- 3.1 x = - 2 - 3.1$

Simplify the arithmetic:

$- 3.1 x = - 5.1$

Divide both sides by :

$\frac{(- 3.1 x)}{- 3.1} = \frac{- 5.1}{- 3.1}$

Cancel out the negatives:

$\frac{3.1 x}{3.1} = \frac{- 5.1}{- 3.1}$

Simplify the arithmetic:

$x = \frac{- 5.1}{- 3.1}$

Cancel out the negatives:

$x = \frac{5.1}{3.1}$

Simplify the arithmetic:

$x = 1.6452$

3. List the solutions

$x = - 1, 1.645$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - x + 3.1 |$
$y = | - 2.1 x + 2 |$
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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