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Solution - Absolute value equations

Exact form:

x = 1, \frac{1}{9}

x=1 , \frac{1}{9}

Decimal form:

x = 1, 0.111

x=1 , 0.111

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
$| 40 x | = | 50 x - 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 40 x \| = \| 50 x - 10 \|$
$x = + y$	$(40 x) = (50 x - 10)$
$x = - y$	$(40 x) = - (50 x - 10)$
$+ x = y$	$(40 x) = (50 x - 10)$
$- x = y$	$- (40 x) = (50 x - 10)$

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:

$\| x \| = \| y \|$	$\| 40 x \| = \| 50 x - 10 \|$
$x = + y$ , $+ x = y$	$(40 x) = (50 x - 10)$
$x = - y$ , $- x = y$	$(40 x) = - (50 x - 10)$

2. Solve the two equations for x

8 additional steps

$40 x = (50 x - 10)$

Subtract from both sides:

$(40 x) - 50 x = (50 x - 10) - 50 x$

Simplify the arithmetic:

$- 10 x = (50 x - 10) - 50 x$

Group like terms:

$- 10 x = (50 x - 50 x) - 10$

Simplify the arithmetic:

$- 10 x = - 10$

Divide both sides by :

$\frac{(- 10 x)}{- 10} = \frac{- 10}{- 10}$

Cancel out the negatives:

$\frac{10 x}{10} = \frac{- 10}{- 10}$

Simplify the fraction:

$x = \frac{- 10}{- 10}$

Cancel out the negatives:

$x = \frac{10}{10}$

Simplify the fraction:

$x = 1$

8 additional steps

$40 x = - (50 x - 10)$

Expand the parentheses:

$40 x = - 50 x + 10$

Add to both sides:

$(40 x) + 50 x = (- 50 x + 10) + 50 x$

Simplify the arithmetic:

$90 x = (- 50 x + 10) + 50 x$

Group like terms:

$90 x = (- 50 x + 50 x) + 10$

Simplify the arithmetic:

$90 x = 10$

Divide both sides by :

$\frac{(90 x)}{90} = \frac{10}{90}$

Simplify the fraction:

$x = \frac{10}{90}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 10)}{(9 \cdot 10)}$

Factor out and cancel the greatest common factor:

$x = \frac{1}{9}$

3. List the solutions

$x = 1, \frac{1}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 40 x |$
$y = | 50 x - 10 |$
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 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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