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Solution - Absolute value equations

Exact form:

x = 5

x=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
$3 | \frac{1}{3} x - 2 | = | - x + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| \frac{1}{3} x - 2 \| = \| - x + 4 \|$
$x = + y$	$3 (\frac{1}{3} x - 2) = (- x + 4)$
$x = - y$	$3 (\frac{1}{3} x - 2) = - (- x + 4)$
$+ x = y$	$3 (\frac{1}{3} x - 2) = (- x + 4)$
$- x = y$	$3 (- (\frac{1}{3} x - 2)) = (- x + 4)$

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:

$\| x \| = \| y \|$	$3 \| \frac{1}{3} x - 2 \| = \| - x + 4 \|$
$x = + y$ , $+ x = y$	$3 (\frac{1}{3} x - 2) = (- x + 4)$
$x = - y$ , $- x = y$	$3 (\frac{1}{3} x - 2) = - (- x + 4)$

2. Solve the two equations for x

15 additional steps

$3 \cdot (\frac{1}{3} x - 2) = (- x + 4)$

Expand the parentheses:

$3 \cdot \frac{1}{3} x + 3 \cdot - 2 = (- x + 4)$

Multiply the coefficients:

$\frac{(3 \cdot 1)}{3} x + 3 \cdot - 2 = (- x + 4)$

Simplify the arithmetic:

$\frac{(3 \cdot 1)}{3} x - 6 = (- x + 4)$

Simplify the fraction:

$x - 6 = (- x + 4)$

Add to both sides:

$(x - 6) + x = (- x + 4) + x$

Group like terms:

$(x + x) - 6 = (- x + 4) + x$

Simplify the arithmetic:

$2 x - 6 = (- x + 4) + x$

Group like terms:

$2 x - 6 = (- x + x) + 4$

Simplify the arithmetic:

$2 x - 6 = 4$

Add to both sides:

$(2 x - 6) + 6 = 4 + 6$

Simplify the arithmetic:

$2 x = 4 + 6$

Simplify the arithmetic:

$2 x = 10$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{10}{2}$

Simplify the fraction:

$x = \frac{10}{2}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = 5$

10 additional steps

$3 \cdot (\frac{1}{3} x - 2) = - (- x + 4)$

Expand the parentheses:

$3 \cdot \frac{1}{3} x + 3 \cdot - 2 = - (- x + 4)$

Multiply the coefficients:

$\frac{(3 \cdot 1)}{3} x + 3 \cdot - 2 = - (- x + 4)$

Simplify the arithmetic:

$\frac{(3 \cdot 1)}{3} x - 6 = - (- x + 4)$

Simplify the fraction:

$x - 6 = - (- x + 4)$

Expand the parentheses:

$x - 6 = x - 4$

Subtract from both sides:

$(x - 6) - x = (x - 4) - x$

Group like terms:

$(x - x) - 6 = (x - 4) - x$

Simplify the arithmetic:

$- 6 = (x - 4) - x$

Group like terms:

$- 6 = (x - x) - 4$

Simplify the arithmetic:

$- 6 = - 4$

The statement is false:

$- 6 = - 4$

The equation is false so it has no solution.

3. List the solutions

$x = 5$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | \frac{1}{3} x - 2 |$
$y = | - x + 4 |$
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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