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Solution - Absolute value equations

Exact form:

y = \frac{5}{2}

y=\frac{5}{2}

Mixed number form:

y = 2 \frac{1}{2}

y=2\frac{1}{2}

Decimal form:

y = 2.5

y=2.5

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
$| - 2 y + 4 | = | 2 y - 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 y + 4 \| = \| 2 y - 6 \|$
$x = + y$	$(- 2 y + 4) = (2 y - 6)$
$x = - y$	$(- 2 y + 4) = - (2 y - 6)$
$+ x = y$	$(- 2 y + 4) = (2 y - 6)$
$- x = y$	$- (- 2 y + 4) = (2 y - 6)$

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:

$\| x \| = \| y \|$	$\| - 2 y + 4 \| = \| 2 y - 6 \|$
$x = + y$ , $+ x = y$	$(- 2 y + 4) = (2 y - 6)$
$x = - y$ , $- x = y$	$(- 2 y + 4) = - (2 y - 6)$

2. Solve the two equations for y

13 additional steps

$(- 2 y + 4) = (2 y - 6)$

Subtract from both sides:

$(- 2 y + 4) - 2 y = (2 y - 6) - 2 y$

Group like terms:

$(- 2 y - 2 y) + 4 = (2 y - 6) - 2 y$

Simplify the arithmetic:

$- 4 y + 4 = (2 y - 6) - 2 y$

Group like terms:

$- 4 y + 4 = (2 y - 2 y) - 6$

Simplify the arithmetic:

$- 4 y + 4 = - 6$

Subtract from both sides:

$(- 4 y + 4) - 4 = - 6 - 4$

Simplify the arithmetic:

$- 4 y = - 6 - 4$

Simplify the arithmetic:

$- 4 y = - 10$

Divide both sides by :

$\frac{(- 4 y)}{- 4} = \frac{- 10}{- 4}$

Cancel out the negatives:

$\frac{4 y}{4} = \frac{- 10}{- 4}$

Simplify the fraction:

$y = \frac{- 10}{- 4}$

Cancel out the negatives:

$y = \frac{10}{4}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(5 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$y = \frac{5}{2}$

6 additional steps

$(- 2 y + 4) = - (2 y - 6)$

Expand the parentheses:

$(- 2 y + 4) = - 2 y + 6$

Add to both sides:

$(- 2 y + 4) + 2 y = (- 2 y + 6) + 2 y$

Group like terms:

$(- 2 y + 2 y) + 4 = (- 2 y + 6) + 2 y$

Simplify the arithmetic:

$4 = (- 2 y + 6) + 2 y$

Group like terms:

$4 = (- 2 y + 2 y) + 6$

Simplify the arithmetic:

$4 = 6$

The statement is false:

$4 = 6$

The equation is false so it has no solution.

3. List the solutions

$y = \frac{5}{2}$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 2 y + 4 |$
$y = | 2 y - 6 |$
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 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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