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Solution - Absolute value equations

Exact form:

x = 2, \frac{1}{9}

x=2 , \frac{1}{9}

Decimal form:

x = 2, 0.111

x=2 , 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
$| 10 x - 3 | = | 8 x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 10 x - 3 \| = \| 8 x + 1 \|$
$x = + y$	$(10 x - 3) = (8 x + 1)$
$x = - y$	$(10 x - 3) = - (8 x + 1)$
$+ x = y$	$(10 x - 3) = (8 x + 1)$
$- x = y$	$- (10 x - 3) = (8 x + 1)$

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:

$\| x \| = \| y \|$	$\| 10 x - 3 \| = \| 8 x + 1 \|$
$x = + y$ , $+ x = y$	$(10 x - 3) = (8 x + 1)$
$x = - y$ , $- x = y$	$(10 x - 3) = - (8 x + 1)$

2. Solve the two equations for x

11 additional steps

$(10 x - 3) = (8 x + 1)$

Subtract from both sides:

$(10 x - 3) - 8 x = (8 x + 1) - 8 x$

Group like terms:

$(10 x - 8 x) - 3 = (8 x + 1) - 8 x$

Simplify the arithmetic:

$2 x - 3 = (8 x + 1) - 8 x$

Group like terms:

$2 x - 3 = (8 x - 8 x) + 1$

Simplify the arithmetic:

$2 x - 3 = 1$

Add to both sides:

$(2 x - 3) + 3 = 1 + 3$

Simplify the arithmetic:

$2 x = 1 + 3$

Simplify the arithmetic:

$2 x = 4$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{4}{2}$

Simplify the fraction:

$x = \frac{4}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 2$

12 additional steps

$(10 x - 3) = - (8 x + 1)$

Expand the parentheses:

$(10 x - 3) = - 8 x - 1$

Add to both sides:

$(10 x - 3) + 8 x = (- 8 x - 1) + 8 x$

Group like terms:

$(10 x + 8 x) - 3 = (- 8 x - 1) + 8 x$

Simplify the arithmetic:

$18 x - 3 = (- 8 x - 1) + 8 x$

Group like terms:

$18 x - 3 = (- 8 x + 8 x) - 1$

Simplify the arithmetic:

$18 x - 3 = - 1$

Add to both sides:

$(18 x - 3) + 3 = - 1 + 3$

Simplify the arithmetic:

$18 x = - 1 + 3$

Simplify the arithmetic:

$18 x = 2$

Divide both sides by :

$\frac{(18 x)}{18} = \frac{2}{18}$

Simplify the fraction:

$x = \frac{2}{18}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 2)}{(9 \cdot 2)}$

Factor out and cancel the greatest common factor:

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

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | 10 x - 3 |$
$y = | 8 x + 1 |$
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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