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Solution - Absolute value equations

Exact form:

y = 3, \frac{7}{4}

y=3 , \frac{7}{4}

Mixed number form:

y = 3, 1 \frac{3}{4}

y=3 , 1\frac{3}{4}

Decimal form:

y = 3, 1.75

y=3 , 1.75

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
$| - 3 y + 4 | = | - 5 y + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 y + 4 \| = \| - 5 y + 10 \|$
$x = + y$	$(- 3 y + 4) = (- 5 y + 10)$
$x = - y$	$(- 3 y + 4) = - (- 5 y + 10)$
$+ x = y$	$(- 3 y + 4) = (- 5 y + 10)$
$- x = y$	$- (- 3 y + 4) = (- 5 y + 10)$

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:

$\| x \| = \| y \|$	$\| - 3 y + 4 \| = \| - 5 y + 10 \|$
$x = + y$ , $+ x = y$	$(- 3 y + 4) = (- 5 y + 10)$
$x = - y$ , $- x = y$	$(- 3 y + 4) = - (- 5 y + 10)$

2. Solve the two equations for y

11 additional steps

$(- 3 y + 4) = (- 5 y + 10)$

Add to both sides:

$(- 3 y + 4) + 5 y = (- 5 y + 10) + 5 y$

Group like terms:

$(- 3 y + 5 y) + 4 = (- 5 y + 10) + 5 y$

Simplify the arithmetic:

$2 y + 4 = (- 5 y + 10) + 5 y$

Group like terms:

$2 y + 4 = (- 5 y + 5 y) + 10$

Simplify the arithmetic:

$2 y + 4 = 10$

Subtract from both sides:

$(2 y + 4) - 4 = 10 - 4$

Simplify the arithmetic:

$2 y = 10 - 4$

Simplify the arithmetic:

$2 y = 6$

Divide both sides by :

$\frac{(2 y)}{2} = \frac{6}{2}$

Simplify the fraction:

$y = \frac{6}{2}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(3 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$y = 3$

14 additional steps

$(- 3 y + 4) = - (- 5 y + 10)$

Expand the parentheses:

$(- 3 y + 4) = 5 y - 10$

Subtract from both sides:

$(- 3 y + 4) - 5 y = (5 y - 10) - 5 y$

Group like terms:

$(- 3 y - 5 y) + 4 = (5 y - 10) - 5 y$

Simplify the arithmetic:

$- 8 y + 4 = (5 y - 10) - 5 y$

Group like terms:

$- 8 y + 4 = (5 y - 5 y) - 10$

Simplify the arithmetic:

$- 8 y + 4 = - 10$

Subtract from both sides:

$(- 8 y + 4) - 4 = - 10 - 4$

Simplify the arithmetic:

$- 8 y = - 10 - 4$

Simplify the arithmetic:

$- 8 y = - 14$

Divide both sides by :

$\frac{(- 8 y)}{- 8} = \frac{- 14}{- 8}$

Cancel out the negatives:

$\frac{8 y}{8} = \frac{- 14}{- 8}$

Simplify the fraction:

$y = \frac{- 14}{- 8}$

Cancel out the negatives:

$y = \frac{14}{8}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(7 \cdot 2)}{(4 \cdot 2)}$

Factor out and cancel the greatest common factor:

$y = \frac{7}{4}$

3. List the solutions

$y = 3, \frac{7}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 3 y + 4 |$
$y = | - 5 y + 10 |$
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 y

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 y

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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