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Solution - Absolute value equations

Exact form:

x = \frac{41}{6}, \frac{69}{4}

x=\frac{41}{6} , \frac{69}{4}

Mixed number form:

x = 6 \frac{5}{6}, 17 \frac{1}{4}

x=6\frac{5}{6} , 17\frac{1}{4}

Decimal form:

x = 6.833, 17.25

x=6.833 , 17.25

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
$5 | x - 11 | = | - x - 14 |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| x - 11 \| = \| - x - 14 \|$
$x = + y$	$5 (x - 11) = (- x - 14)$
$x = - y$	$5 (x - 11) = - (- x - 14)$
$+ x = y$	$5 (x - 11) = (- x - 14)$
$- x = y$	$5 (- (x - 11)) = (- x - 14)$

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:

$\| x \| = \| y \|$	$5 \| x - 11 \| = \| - x - 14 \|$
$x = + y$ , $+ x = y$	$5 (x - 11) = (- x - 14)$
$x = - y$ , $- x = y$	$5 (x - 11) = - (- x - 14)$

2. Solve the two equations for x

11 additional steps

$5 \cdot (x - 11) = (- x - 14)$

Expand the parentheses:

$5 x + 5 \cdot - 11 = (- x - 14)$

Simplify the arithmetic:

$5 x - 55 = (- x - 14)$

Add to both sides:

$(5 x - 55) + x = (- x - 14) + x$

Group like terms:

$(5 x + x) - 55 = (- x - 14) + x$

Simplify the arithmetic:

$6 x - 55 = (- x - 14) + x$

Group like terms:

$6 x - 55 = (- x + x) - 14$

Simplify the arithmetic:

$6 x - 55 = - 14$

Add to both sides:

$(6 x - 55) + 55 = - 14 + 55$

Simplify the arithmetic:

$6 x = - 14 + 55$

Simplify the arithmetic:

$6 x = 41$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{41}{6}$

Simplify the fraction:

$x = \frac{41}{6}$

12 additional steps

$5 \cdot (x - 11) = - (- x - 14)$

Expand the parentheses:

$5 x + 5 \cdot - 11 = - (- x - 14)$

Simplify the arithmetic:

$5 x - 55 = - (- x - 14)$

Expand the parentheses:

$5 x - 55 = x + 14$

Subtract from both sides:

$(5 x - 55) - x = (x + 14) - x$

Group like terms:

$(5 x - x) - 55 = (x + 14) - x$

Simplify the arithmetic:

$4 x - 55 = (x + 14) - x$

Group like terms:

$4 x - 55 = (x - x) + 14$

Simplify the arithmetic:

$4 x - 55 = 14$

Add to both sides:

$(4 x - 55) + 55 = 14 + 55$

Simplify the arithmetic:

$4 x = 14 + 55$

Simplify the arithmetic:

$4 x = 69$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{69}{4}$

Simplify the fraction:

$x = \frac{69}{4}$

3. List the solutions

$x = \frac{41}{6}, \frac{69}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 5 | x - 11 |$
$y = | - x - 14 |$
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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