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Solution - Absolute value equations

Exact form:

x = 40, \frac{80}{173}

x=40 , \frac{80}{173}

Decimal form:

x = 40, 0.462

x=40 , 0.462

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
$| 98 x - 500 | = | 75 x + 420 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 98 x - 500 \| = \| 75 x + 420 \|$
$x = + y$	$(98 x - 500) = (75 x + 420)$
$x = - y$	$(98 x - 500) = - (75 x + 420)$
$+ x = y$	$(98 x - 500) = (75 x + 420)$
$- x = y$	$- (98 x - 500) = (75 x + 420)$

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:

$\| x \| = \| y \|$	$\| 98 x - 500 \| = \| 75 x + 420 \|$
$x = + y$ , $+ x = y$	$(98 x - 500) = (75 x + 420)$
$x = - y$ , $- x = y$	$(98 x - 500) = - (75 x + 420)$

2. Solve the two equations for x

11 additional steps

$(98 x - 500) = (75 x + 420)$

Subtract from both sides:

$(98 x - 500) - 75 x = (75 x + 420) - 75 x$

Group like terms:

$(98 x - 75 x) - 500 = (75 x + 420) - 75 x$

Simplify the arithmetic:

$23 x - 500 = (75 x + 420) - 75 x$

Group like terms:

$23 x - 500 = (75 x - 75 x) + 420$

Simplify the arithmetic:

$23 x - 500 = 420$

Add to both sides:

$(23 x - 500) + 500 = 420 + 500$

Simplify the arithmetic:

$23 x = 420 + 500$

Simplify the arithmetic:

$23 x = 920$

Divide both sides by :

$\frac{(23 x)}{23} = \frac{920}{23}$

Simplify the fraction:

$x = \frac{920}{23}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(40 \cdot 23)}{(1 \cdot 23)}$

Factor out and cancel the greatest common factor:

$x = 40$

10 additional steps

$(98 x - 500) = - (75 x + 420)$

Expand the parentheses:

$(98 x - 500) = - 75 x - 420$

Add to both sides:

$(98 x - 500) + 75 x = (- 75 x - 420) + 75 x$

Group like terms:

$(98 x + 75 x) - 500 = (- 75 x - 420) + 75 x$

Simplify the arithmetic:

$173 x - 500 = (- 75 x - 420) + 75 x$

Group like terms:

$173 x - 500 = (- 75 x + 75 x) - 420$

Simplify the arithmetic:

$173 x - 500 = - 420$

Add to both sides:

$(173 x - 500) + 500 = - 420 + 500$

Simplify the arithmetic:

$173 x = - 420 + 500$

Simplify the arithmetic:

$173 x = 80$

Divide both sides by :

$\frac{(173 x)}{173} = \frac{80}{173}$

Simplify the fraction:

$x = \frac{80}{173}$

3. List the solutions

$x = 40, \frac{80}{173}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 98 x - 500 |$
$y = | 75 x + 420 |$
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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