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Solution - Absolute value equations

Exact form:

u = - 2, \frac{2}{3}

u=-2 , \frac{2}{3}

Decimal form:

u = - 2, 0.667

u=-2 , 0.667

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
$| 4 u | = | 2 u - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 u \| = \| 2 u - 4 \|$
$x = + y$	$(4 u) = (2 u - 4)$
$x = - y$	$(4 u) = - (2 u - 4)$
$+ x = y$	$(4 u) = (2 u - 4)$
$- x = y$	$- (4 u) = (2 u - 4)$

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:

$\| x \| = \| y \|$	$\| 4 u \| = \| 2 u - 4 \|$
$x = + y$ , $+ x = y$	$(4 u) = (2 u - 4)$
$x = - y$ , $- x = y$	$(4 u) = - (2 u - 4)$

2. Solve the two equations for u

7 additional steps

$4 u = (2 u - 4)$

Subtract from both sides:

$(4 u) - 2 u = (2 u - 4) - 2 u$

Simplify the arithmetic:

$2 u = (2 u - 4) - 2 u$

Group like terms:

$2 u = (2 u - 2 u) - 4$

Simplify the arithmetic:

$2 u = - 4$

Divide both sides by :

$\frac{(2 u)}{2} = \frac{- 4}{2}$

Simplify the fraction:

$u = \frac{- 4}{2}$

Find the greatest common factor of the numerator and denominator:

$u = \frac{(- 2 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$u = - 2$

8 additional steps

$4 u = - (2 u - 4)$

Expand the parentheses:

$4 u = - 2 u + 4$

Add to both sides:

$(4 u) + 2 u = (- 2 u + 4) + 2 u$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$6 u = 4$

Divide both sides by :

$\frac{(6 u)}{6} = \frac{4}{6}$

Simplify the fraction:

$u = \frac{4}{6}$

Find the greatest common factor of the numerator and denominator:

$u = \frac{(2 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$u = \frac{2}{3}$

3. List the solutions

$u = - 2, \frac{2}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 u |$
$y = | 2 u - 4 |$
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 u

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 u

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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