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Solution - Absolute value equations

Exact form:

x = \frac{8}{11}, 10

x=\frac{8}{11} , 10

Decimal form:

x = 0.727, 10

x=0.727 , 10

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
$- 3 | 2 x - 3 | = | 5 x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$- 3 \| 2 x - 3 \| = \| 5 x + 1 \|$
$x = + y$	$- 3 (2 x - 3) = (5 x + 1)$
$x = - y$	$- 3 (2 x - 3) = - (5 x + 1)$
$+ x = y$	$- 3 (2 x - 3) = (5 x + 1)$
$- x = y$	$- 3 (- (2 x - 3)) = (5 x + 1)$

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:

$\| x \| = \| y \|$	$- 3 \| 2 x - 3 \| = \| 5 x + 1 \|$
$x = + y$ , $+ x = y$	$- 3 (2 x - 3) = (5 x + 1)$
$x = - y$ , $- x = y$	$- 3 (2 x - 3) = - (5 x + 1)$

2. Solve the two equations for x

14 additional steps

$- 3 \cdot (2 x - 3) = (5 x + 1)$

Expand the parentheses:

$- 3 \cdot 2 x - 3 \cdot - 3 = (5 x + 1)$

Multiply the coefficients:

$- 6 x - 3 \cdot - 3 = (5 x + 1)$

Simplify the arithmetic:

$- 6 x + 9 = (5 x + 1)$

Subtract from both sides:

$(- 6 x + 9) - 5 x = (5 x + 1) - 5 x$

Group like terms:

$(- 6 x - 5 x) + 9 = (5 x + 1) - 5 x$

Simplify the arithmetic:

$- 11 x + 9 = (5 x + 1) - 5 x$

Group like terms:

$- 11 x + 9 = (5 x - 5 x) + 1$

Simplify the arithmetic:

$- 11 x + 9 = 1$

Subtract from both sides:

$(- 11 x + 9) - 9 = 1 - 9$

Simplify the arithmetic:

$- 11 x = 1 - 9$

Simplify the arithmetic:

$- 11 x = - 8$

Divide both sides by :

$\frac{(- 11 x)}{- 11} = \frac{- 8}{- 11}$

Cancel out the negatives:

$\frac{11 x}{11} = \frac{- 8}{- 11}$

Simplify the fraction:

$x = \frac{- 8}{- 11}$

Cancel out the negatives:

$x = \frac{8}{11}$

14 additional steps

$- 3 \cdot (2 x - 3) = - (5 x + 1)$

Expand the parentheses:

$- 3 \cdot 2 x - 3 \cdot - 3 = - (5 x + 1)$

Multiply the coefficients:

$- 6 x - 3 \cdot - 3 = - (5 x + 1)$

Simplify the arithmetic:

$- 6 x + 9 = - (5 x + 1)$

Expand the parentheses:

$- 6 x + 9 = - 5 x - 1$

Add to both sides:

$(- 6 x + 9) + 5 x = (- 5 x - 1) + 5 x$

Group like terms:

$(- 6 x + 5 x) + 9 = (- 5 x - 1) + 5 x$

Simplify the arithmetic:

$- x + 9 = (- 5 x - 1) + 5 x$

Group like terms:

$- x + 9 = (- 5 x + 5 x) - 1$

Simplify the arithmetic:

$- x + 9 = - 1$

Subtract from both sides:

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

Simplify the arithmetic:

$- x = - 1 - 9$

Simplify the arithmetic:

$- x = - 10$

Multiply both sides by :

$- x \cdot - 1 = - 10 \cdot - 1$

Remove the one(s):

$x = - 10 \cdot - 1$

Simplify the arithmetic:

$x = 10$

3. List the solutions

$x = \frac{8}{11}, 10$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = - 3 | 2 x - 3 |$
$y = | 5 x + 1 |$
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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