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Solution - Absolute value equations

Exact form:

x = - \frac{6}{5}, 0

x=-\frac{6}{5} , 0

Mixed number form:

x = - 1 \frac{1}{5}, 0

x=-1\frac{1}{5} , 0

Decimal form:

x = - 1.2, 0

x=-1.2 , 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
$3 | 2 x + 1 | = | x - 3 |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for x

12 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$6 x + 3 \cdot 1 = (x - 3)$

Simplify the arithmetic:

$6 x + 3 = (x - 3)$

Subtract from both sides:

$(6 x + 3) - x = (x - 3) - x$

Group like terms:

$(6 x - x) + 3 = (x - 3) - x$

Simplify the arithmetic:

$5 x + 3 = (x - 3) - x$

Group like terms:

$5 x + 3 = (x - x) - 3$

Simplify the arithmetic:

$5 x + 3 = - 3$

Subtract from both sides:

$(5 x + 3) - 3 = - 3 - 3$

Simplify the arithmetic:

$5 x = - 3 - 3$

Simplify the arithmetic:

$5 x = - 6$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{- 6}{5}$

Simplify the fraction:

$x = \frac{- 6}{5}$

12 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$6 x + 3 \cdot 1 = - (x - 3)$

Simplify the arithmetic:

$6 x + 3 = - (x - 3)$

Expand the parentheses:

$6 x + 3 = - x + 3$

Add to both sides:

$(6 x + 3) + x = (- x + 3) + x$

Group like terms:

$(6 x + x) + 3 = (- x + 3) + x$

Simplify the arithmetic:

$7 x + 3 = (- x + 3) + x$

Group like terms:

$7 x + 3 = (- x + x) + 3$

Simplify the arithmetic:

$7 x + 3 = 3$

Subtract from both sides:

$(7 x + 3) - 3 = 3 - 3$

Simplify the arithmetic:

$7 x = 3 - 3$

Simplify the arithmetic:

$7 x = 0$

Divide both sides by the coefficient:

$x = 0$

3. List the solutions

$x = - \frac{6}{5}, 0$
(2 solution(s))

4. Graph

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