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Solution - Absolute value equations

Exact form:

x = - 2, \frac{4}{7}

x=-2 , \frac{4}{7}

Decimal form:

x = - 2, 0.571

x=-2 , 0.571

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
$3 | 2 x + 1 | = | x - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| 2 x + 1 \| = \| x - 7 \|$
$x = + y$	$3 (2 x + 1) = (x - 7)$
$x = - y$	$3 (2 x + 1) = - (x - 7)$
$+ x = y$	$3 (2 x + 1) = (x - 7)$
$- x = y$	$3 (- (2 x + 1)) = (x - 7)$

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:

$\| x \| = \| y \|$	$3 \| 2 x + 1 \| = \| x - 7 \|$
$x = + y$ , $+ x = y$	$3 (2 x + 1) = (x - 7)$
$x = - y$ , $- x = y$	$3 (2 x + 1) = - (x - 7)$

2. Solve the two equations for x

14 additional steps

$3 \cdot (2 x + 1) = (x - 7)$

Expand the parentheses:

$3 \cdot 2 x + 3 \cdot 1 = (x - 7)$

Multiply the coefficients:

$6 x + 3 \cdot 1 = (x - 7)$

Simplify the arithmetic:

$6 x + 3 = (x - 7)$

Subtract from both sides:

$(6 x + 3) - x = (x - 7) - x$

Group like terms:

$(6 x - x) + 3 = (x - 7) - x$

Simplify the arithmetic:

$5 x + 3 = (x - 7) - x$

Group like terms:

$5 x + 3 = (x - x) - 7$

Simplify the arithmetic:

$5 x + 3 = - 7$

Subtract from both sides:

$(5 x + 3) - 3 = - 7 - 3$

Simplify the arithmetic:

$5 x = - 7 - 3$

Simplify the arithmetic:

$5 x = - 10$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{- 10}{5}$

Simplify the fraction:

$x = \frac{- 10}{5}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = - 2$

13 additional steps

$3 \cdot (2 x + 1) = - (x - 7)$

Expand the parentheses:

$3 \cdot 2 x + 3 \cdot 1 = - (x - 7)$

Multiply the coefficients:

$6 x + 3 \cdot 1 = - (x - 7)$

Simplify the arithmetic:

$6 x + 3 = - (x - 7)$

Expand the parentheses:

$6 x + 3 = - x + 7$

Add to both sides:

$(6 x + 3) + x = (- x + 7) + x$

Group like terms:

$(6 x + x) + 3 = (- x + 7) + x$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$7 x + 3 = 7$

Subtract from both sides:

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

Simplify the arithmetic:

$7 x = 7 - 3$

Simplify the arithmetic:

$7 x = 4$

Divide both sides by :

$\frac{(7 x)}{7} = \frac{4}{7}$

Simplify the fraction:

$x = \frac{4}{7}$

3. List the solutions

$x = - 2, \frac{4}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | 2 x + 1 |$
$y = | x - 7 |$
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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