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Solution - Absolute value equations

Exact form:

x = - \frac{1}{2}, - \frac{1}{10}

x=-\frac{1}{2} , -\frac{1}{10}

Decimal form:

x = - 0.5, - 0.1

x=-0.5 , -0.1

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
$| 2 x | = | 3 x + \frac{1}{2} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x \| = \| 3 x + \frac{1}{2} \|$
$x = + y$	$(2 x) = (3 x + \frac{1}{2})$
$x = - y$	$(2 x) = - (3 x + \frac{1}{2})$
$+ x = y$	$(2 x) = (3 x + \frac{1}{2})$
$- x = y$	$- (2 x) = (3 x + \frac{1}{2})$

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:

$\| x \| = \| y \|$	$\| 2 x \| = \| 3 x + \frac{1}{2} \|$
$x = + y$ , $+ x = y$	$(2 x) = (3 x + \frac{1}{2})$
$x = - y$ , $- x = y$	$(2 x) = - (3 x + \frac{1}{2})$

2. Solve the two equations for x

6 additional steps

$2 x = (3 x + \frac{1}{2})$

Subtract from both sides:

$(2 x) - 3 x = (3 x + \frac{1}{2}) - 3 x$

Simplify the arithmetic:

$- x = (3 x + \frac{1}{2}) - 3 x$

Group like terms:

$- x = (3 x - 3 x) + \frac{1}{2}$

Simplify the arithmetic:

$- x = \frac{1}{2}$

Multiply both sides by :

$- x \cdot - 1 = (\frac{1}{2}) \cdot - 1$

Remove the one(s):

$x = (\frac{1}{2}) \cdot - 1$

Remove the one(s):

$x = \frac{- 1}{2}$

8 additional steps

$2 x = - (3 x + \frac{1}{2})$

Expand the parentheses:

$2 x = - 3 x + \frac{- 1}{2}$

Add to both sides:

$(2 x) + 3 x = (- 3 x + \frac{- 1}{2}) + 3 x$

Simplify the arithmetic:

$5 x = (- 3 x + \frac{- 1}{2}) + 3 x$

Group like terms:

$5 x = (- 3 x + 3 x) + \frac{- 1}{2}$

Simplify the arithmetic:

$5 x = \frac{- 1}{2}$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{(\frac{- 1}{2})}{5}$

Simplify the fraction:

$x = \frac{(\frac{- 1}{2})}{5}$

Simplify the arithmetic:

$x = \frac{- 1}{(2 \cdot 5)}$

$x = \frac{- 1}{10}$

3. List the solutions

$x = - \frac{1}{2}, - \frac{1}{10}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x |$
$y = | 3 x + \frac{1}{2} |$
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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