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Solution - Absolute value equations

Exact form:

u = 10, 10

u=10 , 10

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
$| - u + 10 | = | u - 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - u + 10 \| = \| u - 10 \|$
$x = + y$	$(- u + 10) = (u - 10)$
$x = - y$	$(- u + 10) = - (u - 10)$
$+ x = y$	$(- u + 10) = (u - 10)$
$- x = y$	$- (- u + 10) = (u - 10)$

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:

$\| x \| = \| y \|$	$\| - u + 10 \| = \| u - 10 \|$
$x = + y$ , $+ x = y$	$(- u + 10) = (u - 10)$
$x = - y$ , $- x = y$	$(- u + 10) = - (u - 10)$

2. Solve the two equations for u

13 additional steps

$(- u + 10) = (u - 10)$

Subtract from both sides:

$(- u + 10) - u = (u - 10) - u$

Group like terms:

$(- u - u) + 10 = (u - 10) - u$

Simplify the arithmetic:

$- 2 u + 10 = (u - 10) - u$

Group like terms:

$- 2 u + 10 = (u - u) - 10$

Simplify the arithmetic:

$- 2 u + 10 = - 10$

Subtract from both sides:

$(- 2 u + 10) - 10 = - 10 - 10$

Simplify the arithmetic:

$- 2 u = - 10 - 10$

Simplify the arithmetic:

$- 2 u = - 20$

Divide both sides by :

$\frac{(- 2 u)}{- 2} = \frac{- 20}{- 2}$

Cancel out the negatives:

$\frac{2 u}{2} = \frac{- 20}{- 2}$

Simplify the fraction:

$u = \frac{- 20}{- 2}$

Cancel out the negatives:

$u = \frac{20}{2}$

Find the greatest common factor of the numerator and denominator:

$u = \frac{(10 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$u = 10$

5 additional steps

$(- u + 10) = - (u - 10)$

Expand the parentheses:

$(- u + 10) = - u + 10$

Add to both sides:

$(- u + 10) + u = (- u + 10) + u$

Group like terms:

$(- u + u) + 10 = (- u + 10) + u$

Simplify the arithmetic:

$10 = (- u + 10) + u$

Group like terms:

$10 = (- u + u) + 10$

Simplify the arithmetic:

$10 = 10$

3. List the solutions

$u = 10, 10$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - u + 10 |$
$y = | u - 10 |$
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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