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Solution - Absolute value equations

Exact form:

u = \frac{9}{17}, 3

u=\frac{9}{17} , 3

Decimal form:

u = 0.529, 3

u=0.529 , 3

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
$| 7 u | = | - 10 u + 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 7 u \| = \| - 10 u + 9 \|$
$x = + y$	$(7 u) = (- 10 u + 9)$
$x = - y$	$(7 u) = - (- 10 u + 9)$
$+ x = y$	$(7 u) = (- 10 u + 9)$
$- x = y$	$- (7 u) = (- 10 u + 9)$

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:

$\| x \| = \| y \|$	$\| 7 u \| = \| - 10 u + 9 \|$
$x = + y$ , $+ x = y$	$(7 u) = (- 10 u + 9)$
$x = - y$ , $- x = y$	$(7 u) = - (- 10 u + 9)$

2. Solve the two equations for u

5 additional steps

$7 u = (- 10 u + 9)$

Add to both sides:

$(7 u) + 10 u = (- 10 u + 9) + 10 u$

Simplify the arithmetic:

$17 u = (- 10 u + 9) + 10 u$

Group like terms:

$17 u = (- 10 u + 10 u) + 9$

Simplify the arithmetic:

$17 u = 9$

Divide both sides by :

$\frac{(17 u)}{17} = \frac{9}{17}$

Simplify the fraction:

$u = \frac{9}{17}$

10 additional steps

$7 u = - (- 10 u + 9)$

Expand the parentheses:

$7 u = 10 u - 9$

Subtract from both sides:

$(7 u) - 10 u = (10 u - 9) - 10 u$

Simplify the arithmetic:

$- 3 u = (10 u - 9) - 10 u$

Group like terms:

$- 3 u = (10 u - 10 u) - 9$

Simplify the arithmetic:

$- 3 u = - 9$

Divide both sides by :

$\frac{(- 3 u)}{- 3} = \frac{- 9}{- 3}$

Cancel out the negatives:

$\frac{3 u}{3} = \frac{- 9}{- 3}$

Simplify the fraction:

$u = \frac{- 9}{- 3}$

Cancel out the negatives:

$u = \frac{9}{3}$

Find the greatest common factor of the numerator and denominator:

$u = \frac{(3 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$u = 3$

3. List the solutions

$u = \frac{9}{17}, 3$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 7 u |$
$y = | - 10 u + 9 |$
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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