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Solution - Absolute value equations

Exact form:

x = \frac{3}{14}, \frac{7}{16}

x=\frac{3}{14} , \frac{7}{16}

Decimal form:

x = 0.214, 0.438

x=0.214 , 0.438

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

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

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:

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

2. Solve the two equations for x

12 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$(- x + 2) = - 15 x + 5 \cdot 1$

Simplify the arithmetic:

$(- x + 2) = - 15 x + 5$

Add to both sides:

$(- x + 2) + 15 x = (- 15 x + 5) + 15 x$

Group like terms:

$(- x + 15 x) + 2 = (- 15 x + 5) + 15 x$

Simplify the arithmetic:

$14 x + 2 = (- 15 x + 5) + 15 x$

Group like terms:

$14 x + 2 = (- 15 x + 15 x) + 5$

Simplify the arithmetic:

$14 x + 2 = 5$

Subtract from both sides:

$(14 x + 2) - 2 = 5 - 2$

Simplify the arithmetic:

$14 x = 5 - 2$

Simplify the arithmetic:

$14 x = 3$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{3}{14}$

Simplify the fraction:

$x = \frac{3}{14}$

15 additional steps

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

Expand the parentheses:

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

Expand the parentheses:

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

Multiply the coefficients:

$(- x + 2) = 15 x + 5 \cdot - 1$

Simplify the arithmetic:

$(- x + 2) = 15 x - 5$

Subtract from both sides:

$(- x + 2) - 15 x = (15 x - 5) - 15 x$

Group like terms:

$(- x - 15 x) + 2 = (15 x - 5) - 15 x$

Simplify the arithmetic:

$- 16 x + 2 = (15 x - 5) - 15 x$

Group like terms:

$- 16 x + 2 = (15 x - 15 x) - 5$

Simplify the arithmetic:

$- 16 x + 2 = - 5$

Subtract from both sides:

$(- 16 x + 2) - 2 = - 5 - 2$

Simplify the arithmetic:

$- 16 x = - 5 - 2$

Simplify the arithmetic:

$- 16 x = - 7$

Divide both sides by :

$\frac{(- 16 x)}{- 16} = \frac{- 7}{- 16}$

Cancel out the negatives:

$\frac{16 x}{16} = \frac{- 7}{- 16}$

Simplify the fraction:

$x = \frac{- 7}{- 16}$

Cancel out the negatives:

$x = \frac{7}{16}$

3. List the solutions

$x = \frac{3}{14}, \frac{7}{16}$
(2 solution(s))

4. Graph

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