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Solution - Absolute value equations

Exact form:

x = \frac{14}{11}, \frac{14}{17}

x=\frac{14}{11} , \frac{14}{17}

Mixed number form:

x = 1 \frac{3}{11}, \frac{14}{17}

x=1\frac{3}{11} , \frac{14}{17}

Decimal form:

x = 1.273, 0.824

x=1.273 , 0.824

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
$| 4 x - \frac{4}{1} | = | \frac{6}{7} x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - \frac{4}{1} \| = \| \frac{6}{7} x \|$
$x = + y$	$(4 x - \frac{4}{1}) = (\frac{6}{7} x)$
$x = - y$	$(4 x - \frac{4}{1}) = - (\frac{6}{7} x)$
$+ x = y$	$(4 x - \frac{4}{1}) = (\frac{6}{7} x)$
$- x = y$	$- (4 x - \frac{4}{1}) = (\frac{6}{7} x)$

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:

$\| x \| = \| y \|$	$\| 4 x - \frac{4}{1} \| = \| \frac{6}{7} x \|$
$x = + y$ , $+ x = y$	$(4 x - \frac{4}{1}) = (\frac{6}{7} x)$
$x = - y$ , $- x = y$	$(4 x - \frac{4}{1}) = - (\frac{6}{7} x)$

2. Solve the two equations for x

19 additional steps

$4 x + \frac{- 4}{1} = \frac{6}{7} x$

A variable's value does not change when it is divided by 1, so we can eliminate it:

$4 x - 4 = \frac{6}{7} x$

Subtract from both sides:

$(4 x - 4) - \frac{6}{7} \cdot x = (\frac{6}{7} x) - \frac{6}{7} x$

Group like terms:

$(4 x + \frac{- 6}{7} \cdot x) - 4 = (\frac{6}{7} \cdot x) - \frac{6}{7} x$

Group the coefficients:

$(4 + \frac{- 6}{7}) x - 4 = (\frac{6}{7} \cdot x) - \frac{6}{7} x$

Convert the integer into a fraction:

$(\frac{28}{7} + \frac{- 6}{7}) x - 4 = (\frac{6}{7} \cdot x) - \frac{6}{7} x$

Combine the fractions:

$\frac{(28 - 6)}{7} \cdot x - 4 = (\frac{6}{7} \cdot x) - \frac{6}{7} x$

Combine the numerators:

$\frac{22}{7} \cdot x - 4 = (\frac{6}{7} \cdot x) - \frac{6}{7} x$

Combine the fractions:

$\frac{22}{7} \cdot x - 4 = \frac{(6 - 6)}{7} x$

Combine the numerators:

$\frac{22}{7} \cdot x - 4 = \frac{0}{7} x$

Reduce the zero numerator:

$\frac{22}{7} x - 4 = 0 x$

Simplify the arithmetic:

$\frac{22}{7} x - 4 = 0$

Add to both sides:

$(\frac{22}{7} x - 4) + 4 = 0 + 4$

Simplify the arithmetic:

$\frac{22}{7} x = 0 + 4$

Simplify the arithmetic:

$\frac{22}{7} x = 4$

Multiply both sides by inverse fraction :

$(\frac{22}{7} x) \cdot \frac{7}{22} = 4 \cdot \frac{7}{22}$

Group like terms:

$(\frac{22}{7} \cdot \frac{7}{22}) x = 4 \cdot \frac{7}{22}$

Multiply the coefficients:

$\frac{(22 \cdot 7)}{(7 \cdot 22)} x = 4 \cdot \frac{7}{22}$

Simplify the fraction:

$x = 4 \cdot \frac{7}{22}$

Multiply the fraction(s):

$x = \frac{(4 \cdot 7)}{22}$

Simplify the arithmetic:

$x = \frac{14}{11}$

19 additional steps

$4 x + \frac{- 4}{1} = - (\frac{6}{7} x)$

A variable's value does not change when it is divided by 1, so we can eliminate it:

$4 x - 4 = - (\frac{6}{7} x)$

Add to both sides:

$(4 x - 4) + \frac{6}{7} \cdot x = (\frac{- 6}{7} x) + \frac{6}{7} x$

Group like terms:

$(4 x + \frac{6}{7} \cdot x) - 4 = (\frac{- 6}{7} \cdot x) + \frac{6}{7} x$

Group the coefficients:

$(4 + \frac{6}{7}) x - 4 = (\frac{- 6}{7} \cdot x) + \frac{6}{7} x$

Convert the integer into a fraction:

$(\frac{28}{7} + \frac{6}{7}) x - 4 = (\frac{- 6}{7} \cdot x) + \frac{6}{7} x$

Combine the fractions:

$\frac{(28 + 6)}{7} \cdot x - 4 = (\frac{- 6}{7} \cdot x) + \frac{6}{7} x$

Combine the numerators:

$\frac{34}{7} \cdot x - 4 = (\frac{- 6}{7} \cdot x) + \frac{6}{7} x$

Combine the fractions:

$\frac{34}{7} \cdot x - 4 = \frac{(- 6 + 6)}{7} x$

Combine the numerators:

$\frac{34}{7} \cdot x - 4 = \frac{0}{7} x$

Reduce the zero numerator:

$\frac{34}{7} x - 4 = 0 x$

Simplify the arithmetic:

$\frac{34}{7} x - 4 = 0$

Add to both sides:

$(\frac{34}{7} x - 4) + 4 = 0 + 4$

Simplify the arithmetic:

$\frac{34}{7} x = 0 + 4$

Simplify the arithmetic:

$\frac{34}{7} x = 4$

Multiply both sides by inverse fraction :

$(\frac{34}{7} x) \cdot \frac{7}{34} = 4 \cdot \frac{7}{34}$

Group like terms:

$(\frac{34}{7} \cdot \frac{7}{34}) x = 4 \cdot \frac{7}{34}$

Multiply the coefficients:

$\frac{(34 \cdot 7)}{(7 \cdot 34)} x = 4 \cdot \frac{7}{34}$

Simplify the fraction:

$x = 4 \cdot \frac{7}{34}$

Multiply the fraction(s):

$x = \frac{(4 \cdot 7)}{34}$

Simplify the arithmetic:

$x = \frac{14}{17}$

3. List the solutions

$x = \frac{14}{11}, \frac{14}{17}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - \frac{4}{1} |$
$y = | \frac{6}{7} x |$
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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