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Solution - Absolute value equations

Exact form:

x = 5, \frac{15}{7}

x=5 , \frac{15}{7}

Mixed number form:

x = 5, 2 \frac{1}{7}

x=5 , 2\frac{1}{7}

Decimal form:

x = 5, 2.143

x=5 , 2.143

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
$| \frac{2}{5} x | = | x - 3 |$
without the absolute value bars:

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

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:

$\| x \| = \| y \|$	$\| \frac{2}{5} x \| = \| x - 3 \|$
$x = + y$ , $+ x = y$	$(\frac{2}{5} x) = (x - 3)$
$x = - y$ , $- x = y$	$(\frac{2}{5} x) = - (x - 3)$

2. Solve the two equations for x

15 additional steps

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

Subtract from both sides:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Group like terms:

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

Simplify the arithmetic:

$\frac{- 3}{5} x = - 3$

Multiply both sides by inverse fraction :

$(\frac{- 3}{5} x) \cdot \frac{5}{- 3} = - 3 \cdot \frac{5}{- 3}$

Move the negative sign from the denominator to the numerator:

$\frac{- 3}{5} x \cdot \frac{- 5}{3} = - 3 \cdot \frac{5}{- 3}$

Group like terms:

$(\frac{- 3}{5} \cdot \frac{- 5}{3}) x = - 3 \cdot \frac{5}{- 3}$

Multiply the coefficients:

$\frac{(- 3 \cdot - 5)}{(5 \cdot 3)} x = - 3 \cdot \frac{5}{- 3}$

Simplify the arithmetic:

$1 x = - 3 \cdot \frac{5}{- 3}$

$x = - 3 \cdot \frac{5}{- 3}$

Move the negative sign from the denominator to the numerator:

$x = - 3 \cdot \frac{- 5}{3}$

Multiply the fraction(s):

$x = \frac{(- 3 \cdot - 5)}{3}$

Simplify the arithmetic:

$x = 5$

13 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

$\frac{7}{5} x = (- x + 3) + x$

Group like terms:

$\frac{7}{5} x = (- x + x) + 3$

Simplify the arithmetic:

$\frac{7}{5} x = 3$

Multiply both sides by inverse fraction :

$(\frac{7}{5} x) \cdot \frac{5}{7} = 3 \cdot \frac{5}{7}$

Group like terms:

$(\frac{7}{5} \cdot \frac{5}{7}) x = 3 \cdot \frac{5}{7}$

Multiply the coefficients:

$\frac{(7 \cdot 5)}{(5 \cdot 7)} x = 3 \cdot \frac{5}{7}$

Simplify the fraction:

$x = 3 \cdot \frac{5}{7}$

Multiply the fraction(s):

$x = \frac{(3 \cdot 5)}{7}$

Simplify the arithmetic:

$x = \frac{15}{7}$

3. List the solutions

$x = 5, \frac{15}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{2}{5} x |$
$y = | x - 3 |$
The equation is true where the two lines cross.

How did we do?

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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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