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Solution - Absolute value equations

Exact form:

x = \frac{5}{12}, - \frac{5}{2}

x=\frac{5}{12} , -\frac{5}{2}

Mixed number form:

x = \frac{5}{12}, - 2 \frac{1}{2}

x=\frac{5}{12} , -2\frac{1}{2}

Decimal form:

x = 0.417, - 2.5

x=0.417 , -2.5

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
$5 | - x + 1 | = | 7 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| - x + 1 \| = \| 7 x \|$
$x = + y$	$5 (- x + 1) = (7 x)$
$x = - y$	$5 (- x + 1) = - (7 x)$
$+ x = y$	$5 (- x + 1) = (7 x)$
$- x = y$	$5 (- (- x + 1)) = (7 x)$

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:

$\| x \| = \| y \|$	$5 \| - x + 1 \| = \| 7 x \|$
$x = + y$ , $+ x = y$	$5 (- x + 1) = (7 x)$
$x = - y$ , $- x = y$	$5 (- x + 1) = - (7 x)$

2. Solve the two equations for x

14 additional steps

$5 \cdot (- x + 1) = 7 x$

Expand the parentheses:

$5 \cdot - x + 5 \cdot 1 = 7 x$

Group like terms:

$(5 \cdot - 1) x + 5 \cdot 1 = 7 x$

Multiply the coefficients:

$- 5 x + 5 \cdot 1 = 7 x$

Simplify the arithmetic:

$- 5 x + 5 = 7 x$

Subtract from both sides:

$(- 5 x + 5) - 7 x = (7 x) - 7 x$

Group like terms:

$(- 5 x - 7 x) + 5 = (7 x) - 7 x$

Simplify the arithmetic:

$- 12 x + 5 = (7 x) - 7 x$

Simplify the arithmetic:

$- 12 x + 5 = 0$

Subtract from both sides:

$(- 12 x + 5) - 5 = 0 - 5$

Simplify the arithmetic:

$- 12 x = 0 - 5$

Simplify the arithmetic:

$- 12 x = - 5$

Divide both sides by :

$\frac{(- 12 x)}{- 12} = \frac{- 5}{- 12}$

Cancel out the negatives:

$\frac{12 x}{12} = \frac{- 5}{- 12}$

Simplify the fraction:

$x = \frac{- 5}{- 12}$

Cancel out the negatives:

$x = \frac{5}{12}$

12 additional steps

$5 \cdot (- x + 1) = - (7 x)$

Expand the parentheses:

$5 \cdot - x + 5 \cdot 1 = - (7 x)$

Group like terms:

$(5 \cdot - 1) x + 5 \cdot 1 = - (7 x)$

Multiply the coefficients:

$- 5 x + 5 \cdot 1 = - (7 x)$

Simplify the arithmetic:

$- 5 x + 5 = - (7 x)$

Add to both sides:

$(- 5 x + 5) + 7 x = (- 7 x) + 7 x$

Group like terms:

$(- 5 x + 7 x) + 5 = (- 7 x) + 7 x$

Simplify the arithmetic:

$2 x + 5 = (- 7 x) + 7 x$

Simplify the arithmetic:

$2 x + 5 = 0$

Subtract from both sides:

$(2 x + 5) - 5 = 0 - 5$

Simplify the arithmetic:

$2 x = 0 - 5$

Simplify the arithmetic:

$2 x = - 5$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{- 5}{2}$

Simplify the fraction:

$x = \frac{- 5}{2}$

3. List the solutions

$x = \frac{5}{12}, - \frac{5}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 5 | - x + 1 |$
$y = | 7 x |$
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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