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Solution - Absolute value equations

Exact form:

x = 7, \frac{5}{2}

x=7 , \frac{5}{2}

Mixed number form:

x = 7, 2 \frac{1}{2}

x=7 , 2\frac{1}{2}

Decimal form:

x = 7, 2.5

x=7 , 2.5

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
$3 | x - 4 | = | x + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| x - 4 \| = \| x + 2 \|$
$x = + y$	$3 (x - 4) = (x + 2)$
$x = - y$	$3 (x - 4) = - (x + 2)$
$+ x = y$	$3 (x - 4) = (x + 2)$
$- x = y$	$3 (- (x - 4)) = (x + 2)$

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:

$\| x \| = \| y \|$	$3 \| x - 4 \| = \| x + 2 \|$
$x = + y$ , $+ x = y$	$3 (x - 4) = (x + 2)$
$x = - y$ , $- x = y$	$3 (x - 4) = - (x + 2)$

2. Solve the two equations for x

13 additional steps

$3 \cdot (x - 4) = (x + 2)$

Expand the parentheses:

$3 x + 3 \cdot - 4 = (x + 2)$

Simplify the arithmetic:

$3 x - 12 = (x + 2)$

Subtract from both sides:

$(3 x - 12) - x = (x + 2) - x$

Group like terms:

$(3 x - x) - 12 = (x + 2) - x$

Simplify the arithmetic:

$2 x - 12 = (x + 2) - x$

Group like terms:

$2 x - 12 = (x - x) + 2$

Simplify the arithmetic:

$2 x - 12 = 2$

Add to both sides:

$(2 x - 12) + 12 = 2 + 12$

Simplify the arithmetic:

$2 x = 2 + 12$

Simplify the arithmetic:

$2 x = 14$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{14}{2}$

Simplify the fraction:

$x = \frac{14}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(7 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 7$

14 additional steps

$3 \cdot (x - 4) = - (x + 2)$

Expand the parentheses:

$3 x + 3 \cdot - 4 = - (x + 2)$

Simplify the arithmetic:

$3 x - 12 = - (x + 2)$

Expand the parentheses:

$3 x - 12 = - x - 2$

Add to both sides:

$(3 x - 12) + x = (- x - 2) + x$

Group like terms:

$(3 x + x) - 12 = (- x - 2) + x$

Simplify the arithmetic:

$4 x - 12 = (- x - 2) + x$

Group like terms:

$4 x - 12 = (- x + x) - 2$

Simplify the arithmetic:

$4 x - 12 = - 2$

Add to both sides:

$(4 x - 12) + 12 = - 2 + 12$

Simplify the arithmetic:

$4 x = - 2 + 12$

Simplify the arithmetic:

$4 x = 10$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{10}{4}$

Simplify the fraction:

$x = \frac{10}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

$x = 7, \frac{5}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | x - 4 |$
$y = | x + 2 |$
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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