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Solution - Absolute value equations

Exact form:

x = \frac{4}{15}, \frac{4}{3}

x=\frac{4}{15} , \frac{4}{3}

Mixed number form:

x = \frac{4}{15}, 1 \frac{1}{3}

x=\frac{4}{15} , 1\frac{1}{3}

Decimal form:

x = 0.267, 1.333

x=0.267 , 1.333

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

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

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:

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

2. Solve the two equations for x

13 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$5 x + \frac{- 4}{3} = (- 2 x) + 2 x$

Simplify the arithmetic:

$5 x + \frac{- 4}{3} = 0$

Add to both sides:

$(5 x + \frac{- 4}{3}) + \frac{4}{3} = 0 + \frac{4}{3}$

Combine the fractions:

$5 x + \frac{(- 4 + 4)}{3} = 0 + \frac{4}{3}$

Combine the numerators:

$5 x + \frac{0}{3} = 0 + \frac{4}{3}$

Reduce the zero numerator:

$5 x + 0 = 0 + \frac{4}{3}$

Simplify the arithmetic:

$5 x = 0 + \frac{4}{3}$

Simplify the arithmetic:

$5 x = \frac{4}{3}$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{(\frac{4}{3})}{5}$

Simplify the fraction:

$x = \frac{(\frac{4}{3})}{5}$

Simplify the arithmetic:

$x = \frac{4}{(3 \cdot 5)}$

$x = \frac{4}{15}$

10 additional steps

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

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$x + \frac{- 4}{3} = 0$

Add to both sides:

$(x + \frac{- 4}{3}) + \frac{4}{3} = 0 + \frac{4}{3}$

Combine the fractions:

$x + \frac{(- 4 + 4)}{3} = 0 + \frac{4}{3}$

Combine the numerators:

$x + \frac{0}{3} = 0 + \frac{4}{3}$

Reduce the zero numerator:

$x + 0 = 0 + \frac{4}{3}$

Simplify the arithmetic:

$x = 0 + \frac{4}{3}$

Simplify the arithmetic:

$x = \frac{4}{3}$

3. List the solutions

$x = \frac{4}{15}, \frac{4}{3}$
(2 solution(s))

4. Graph

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