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Solution - Absolute value equations

Exact form:

x = \frac{29}{7}, - \frac{9}{13}

x=\frac{29}{7} , -\frac{9}{13}

Mixed number form:

x = 4 \frac{1}{7}, - \frac{9}{13}

x=4\frac{1}{7} , -\frac{9}{13}

Decimal form:

x = 4.143, - 0.692

x=4.143 , -0.692

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
$| 10 x - 10 | = | 3 x + 19 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 10 x - 10 \| = \| 3 x + 19 \|$
$x = + y$	$(10 x - 10) = (3 x + 19)$
$x = - y$	$(10 x - 10) = - (3 x + 19)$
$+ x = y$	$(10 x - 10) = (3 x + 19)$
$- x = y$	$- (10 x - 10) = (3 x + 19)$

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:

$\| x \| = \| y \|$	$\| 10 x - 10 \| = \| 3 x + 19 \|$
$x = + y$ , $+ x = y$	$(10 x - 10) = (3 x + 19)$
$x = - y$ , $- x = y$	$(10 x - 10) = - (3 x + 19)$

2. Solve the two equations for x

9 additional steps

$(10 x - 10) = (3 x + 19)$

Subtract from both sides:

$(10 x - 10) - 3 x = (3 x + 19) - 3 x$

Group like terms:

$(10 x - 3 x) - 10 = (3 x + 19) - 3 x$

Simplify the arithmetic:

$7 x - 10 = (3 x + 19) - 3 x$

Group like terms:

$7 x - 10 = (3 x - 3 x) + 19$

Simplify the arithmetic:

$7 x - 10 = 19$

Add to both sides:

$(7 x - 10) + 10 = 19 + 10$

Simplify the arithmetic:

$7 x = 19 + 10$

Simplify the arithmetic:

$7 x = 29$

Divide both sides by :

$\frac{(7 x)}{7} = \frac{29}{7}$

Simplify the fraction:

$x = \frac{29}{7}$

10 additional steps

$(10 x - 10) = - (3 x + 19)$

Expand the parentheses:

$(10 x - 10) = - 3 x - 19$

Add to both sides:

$(10 x - 10) + 3 x = (- 3 x - 19) + 3 x$

Group like terms:

$(10 x + 3 x) - 10 = (- 3 x - 19) + 3 x$

Simplify the arithmetic:

$13 x - 10 = (- 3 x - 19) + 3 x$

Group like terms:

$13 x - 10 = (- 3 x + 3 x) - 19$

Simplify the arithmetic:

$13 x - 10 = - 19$

Add to both sides:

$(13 x - 10) + 10 = - 19 + 10$

Simplify the arithmetic:

$13 x = - 19 + 10$

Simplify the arithmetic:

$13 x = - 9$

Divide both sides by :

$\frac{(13 x)}{13} = \frac{- 9}{13}$

Simplify the fraction:

$x = \frac{- 9}{13}$

3. List the solutions

$x = \frac{29}{7}, - \frac{9}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 10 x - 10 |$
$y = | 3 x + 19 |$
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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