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Solution - Absolute value equations

Exact form:

x = \frac{7}{3}, \frac{19}{11}

x=\frac{7}{3} , \frac{19}{11}

Mixed number form:

x = 2 \frac{1}{3}, 1 \frac{8}{11}

x=2\frac{1}{3} , 1\frac{8}{11}

Decimal form:

x = 2.333, 1.727

x=2.333 , 1.727

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
$- 5 | 2 x - 4 | = - | x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$- 5 \| 2 x - 4 \| = - \| x + 1 \|$
$x = + y$	$- 5 (2 x - 4) = - (x + 1)$
$x = - y$	$- 5 (2 x - 4) = - (- (x + 1))$
$+ x = y$	$- 5 (2 x - 4) = - (x + 1)$
$- x = y$	$- 5 (- (2 x - 4)) = - (x + 1)$

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:

$\| x \| = \| y \|$	$- 5 \| 2 x - 4 \| = - \| x + 1 \|$
$x = + y$ , $+ x = y$	$- 5 (2 x - 4) = - (x + 1)$
$x = - y$ , $- x = y$	$- 5 (2 x - 4) = - (- (x + 1))$

2. Solve the two equations for x

17 additional steps

$- 5 \cdot (2 x - 4) = - (x + 1)$

Expand the parentheses:

$- 5 \cdot 2 x - 5 \cdot - 4 = - (x + 1)$

Multiply the coefficients:

$- 10 x - 5 \cdot - 4 = - (x + 1)$

Simplify the arithmetic:

$- 10 x + 20 = - (x + 1)$

Expand the parentheses:

$- 10 x + 20 = - x - 1$

Add to both sides:

$(- 10 x + 20) + x = (- x - 1) + x$

Group like terms:

$(- 10 x + x) + 20 = (- x - 1) + x$

Simplify the arithmetic:

$- 9 x + 20 = (- x - 1) + x$

Group like terms:

$- 9 x + 20 = (- x + x) - 1$

Simplify the arithmetic:

$- 9 x + 20 = - 1$

Subtract from both sides:

$(- 9 x + 20) - 20 = - 1 - 20$

Simplify the arithmetic:

$- 9 x = - 1 - 20$

Simplify the arithmetic:

$- 9 x = - 21$

Divide both sides by :

$\frac{(- 9 x)}{- 9} = \frac{- 21}{- 9}$

Cancel out the negatives:

$\frac{9 x}{9} = \frac{- 21}{- 9}$

Simplify the fraction:

$x = \frac{- 21}{- 9}$

Cancel out the negatives:

$x = \frac{21}{9}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(7 \cdot 3)}{(3 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{7}{3}$

15 additional steps

$- 5 \cdot (2 x - 4) = - (- (x + 1))$

Expand the parentheses:

$- 5 \cdot 2 x - 5 \cdot - 4 = - (- (x + 1))$

Multiply the coefficients:

$- 10 x - 5 \cdot - 4 = - (- (x + 1))$

Simplify the arithmetic:

$- 10 x + 20 = - (- (x + 1))$

Resolve the double minus:

$- 10 x + 20 = x + 1$

Subtract from both sides:

$(- 10 x + 20) - x = (x + 1) - x$

Group like terms:

$(- 10 x - x) + 20 = (x + 1) - x$

Simplify the arithmetic:

$- 11 x + 20 = (x + 1) - x$

Group like terms:

$- 11 x + 20 = (x - x) + 1$

Simplify the arithmetic:

$- 11 x + 20 = 1$

Subtract from both sides:

$(- 11 x + 20) - 20 = 1 - 20$

Simplify the arithmetic:

$- 11 x = 1 - 20$

Simplify the arithmetic:

$- 11 x = - 19$

Divide both sides by :

$\frac{(- 11 x)}{- 11} = \frac{- 19}{- 11}$

Cancel out the negatives:

$\frac{11 x}{11} = \frac{- 19}{- 11}$

Simplify the fraction:

$x = \frac{- 19}{- 11}$

Cancel out the negatives:

$x = \frac{19}{11}$

3. List the solutions

$x = \frac{7}{3}, \frac{19}{11}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = - 5 | 2 x - 4 |$
$y = - | x + 1 |$
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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