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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = - 0.111, 1

x=-0.111 , 1

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
$5 | 2 x - 1 | = | x - 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| 2 x - 1 \| = \| x - 6 \|$
$x = + y$	$5 (2 x - 1) = (x - 6)$
$x = - y$	$5 (2 x - 1) = - (x - 6)$
$+ x = y$	$5 (2 x - 1) = (x - 6)$
$- x = y$	$5 (- (2 x - 1)) = (x - 6)$

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:

$\| x \| = \| y \|$	$5 \| 2 x - 1 \| = \| x - 6 \|$
$x = + y$ , $+ x = y$	$5 (2 x - 1) = (x - 6)$
$x = - y$ , $- x = y$	$5 (2 x - 1) = - (x - 6)$

2. Solve the two equations for x

12 additional steps

$5 \cdot (2 x - 1) = (x - 6)$

Expand the parentheses:

$5 \cdot 2 x + 5 \cdot - 1 = (x - 6)$

Multiply the coefficients:

$10 x + 5 \cdot - 1 = (x - 6)$

Simplify the arithmetic:

$10 x - 5 = (x - 6)$

Subtract from both sides:

$(10 x - 5) - x = (x - 6) - x$

Group like terms:

$(10 x - x) - 5 = (x - 6) - x$

Simplify the arithmetic:

$9 x - 5 = (x - 6) - x$

Group like terms:

$9 x - 5 = (x - x) - 6$

Simplify the arithmetic:

$9 x - 5 = - 6$

Add to both sides:

$(9 x - 5) + 5 = - 6 + 5$

Simplify the arithmetic:

$9 x = - 6 + 5$

Simplify the arithmetic:

$9 x = - 1$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{- 1}{9}$

Simplify the fraction:

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

14 additional steps

$5 \cdot (2 x - 1) = - (x - 6)$

Expand the parentheses:

$5 \cdot 2 x + 5 \cdot - 1 = - (x - 6)$

Multiply the coefficients:

$10 x + 5 \cdot - 1 = - (x - 6)$

Simplify the arithmetic:

$10 x - 5 = - (x - 6)$

Expand the parentheses:

$10 x - 5 = - x + 6$

Add to both sides:

$(10 x - 5) + x = (- x + 6) + x$

Group like terms:

$(10 x + x) - 5 = (- x + 6) + x$

Simplify the arithmetic:

$11 x - 5 = (- x + 6) + x$

Group like terms:

$11 x - 5 = (- x + x) + 6$

Simplify the arithmetic:

$11 x - 5 = 6$

Add to both sides:

$(11 x - 5) + 5 = 6 + 5$

Simplify the arithmetic:

$11 x = 6 + 5$

Simplify the arithmetic:

$11 x = 11$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$x = 1$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = 5 | 2 x - 1 |$
$y = | x - 6 |$
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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