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Solution - Absolute value equations

Exact form:

x = \frac{4}{3}

x=\frac{4}{3}

Mixed number form:

x = 1 \frac{1}{3}

x=1\frac{1}{3}

Decimal form:

x = 1.333

x=1.333

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$3 | x | - | 3 x - 8 | = 0$

Add $| 3 x - 8 |$ to both sides of the equation:

$3 | x | - | 3 x - 8 | + | 3 x - 8 | = | 3 x - 8 |$

Simplify the arithmetic

$3 | x | = | 3 x - 8 |$

2. 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
$3 | x | = | 3 x - 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| x \| = \| 3 x - 8 \|$
$x = + y$	$3 (x) = (3 x - 8)$
$x = - y$	$3 (x) = (- (3 x - 8))$
$+ x = y$	$3 (x) = (3 x - 8)$
$- x = y$	$3 (- (x)) = (3 x - 8)$

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:

$\| x \| = \| y \|$	$3 \| x \| = \| 3 x - 8 \|$
$x = + y$ , $+ x = y$	$3 (x) = (3 x - 8)$
$x = - y$ , $- x = y$	$3 (x) = (- (3 x - 8))$

3. Solve the two equations for x

4 additional steps

$3 x = (3 x - 8)$

Subtract from both sides:

$(3 x) - 3 x = (3 x - 8) - 3 x$

Simplify the arithmetic:

$0 = (3 x - 8) - 3 x$

Group like terms:

$0 = (3 x - 3 x) - 8$

Simplify the arithmetic:

$0 = - 8$

The statement is false:

$0 = - 8$

The equation is false so it has no solution.

8 additional steps

$3 x = - (3 x - 8)$

Expand the parentheses:

$3 x = - 3 x + 8$

Add to both sides:

$(3 x) + 3 x = (- 3 x + 8) + 3 x$

Simplify the arithmetic:

$6 x = (- 3 x + 8) + 3 x$

Group like terms:

$6 x = (- 3 x + 3 x) + 8$

Simplify the arithmetic:

$6 x = 8$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{8}{6}$

Simplify the fraction:

$x = \frac{8}{6}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | x |$
$y = | 3 x - 8 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. Graph

Why learn this

Terms and topics

Related links

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