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Solution - Absolute value equations

Exact form:

y = 4, 16

y=4 , 16

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
$| 2 y - 2 | = | - 3 y + 18 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 y - 2 \| = \| - 3 y + 18 \|$
$x = + y$	$(2 y - 2) = (- 3 y + 18)$
$x = - y$	$(2 y - 2) = - (- 3 y + 18)$
$+ x = y$	$(2 y - 2) = (- 3 y + 18)$
$- x = y$	$- (2 y - 2) = (- 3 y + 18)$

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:

$\| x \| = \| y \|$	$\| 2 y - 2 \| = \| - 3 y + 18 \|$
$x = + y$ , $+ x = y$	$(2 y - 2) = (- 3 y + 18)$
$x = - y$ , $- x = y$	$(2 y - 2) = - (- 3 y + 18)$

2. Solve the two equations for y

11 additional steps

$(2 y - 2) = (- 3 y + 18)$

Add to both sides:

$(2 y - 2) + 3 y = (- 3 y + 18) + 3 y$

Group like terms:

$(2 y + 3 y) - 2 = (- 3 y + 18) + 3 y$

Simplify the arithmetic:

$5 y - 2 = (- 3 y + 18) + 3 y$

Group like terms:

$5 y - 2 = (- 3 y + 3 y) + 18$

Simplify the arithmetic:

$5 y - 2 = 18$

Add to both sides:

$(5 y - 2) + 2 = 18 + 2$

Simplify the arithmetic:

$5 y = 18 + 2$

Simplify the arithmetic:

$5 y = 20$

Divide both sides by :

$\frac{(5 y)}{5} = \frac{20}{5}$

Simplify the fraction:

$y = \frac{20}{5}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(4 \cdot 5)}{(1 \cdot 5)}$

Factor out and cancel the greatest common factor:

$y = 4$

11 additional steps

$(2 y - 2) = - (- 3 y + 18)$

Expand the parentheses:

$(2 y - 2) = 3 y - 18$

Subtract from both sides:

$(2 y - 2) - 3 y = (3 y - 18) - 3 y$

Group like terms:

$(2 y - 3 y) - 2 = (3 y - 18) - 3 y$

Simplify the arithmetic:

$- y - 2 = (3 y - 18) - 3 y$

Group like terms:

$- y - 2 = (3 y - 3 y) - 18$

Simplify the arithmetic:

$- y - 2 = - 18$

Add to both sides:

$(- y - 2) + 2 = - 18 + 2$

Simplify the arithmetic:

$- y = - 18 + 2$

Simplify the arithmetic:

$- y = - 16$

Multiply both sides by :

$- y \cdot - 1 = - 16 \cdot - 1$

Remove the one(s):

$y = - 16 \cdot - 1$

Simplify the arithmetic:

$y = 16$

3. List the solutions

$y = 4, 16$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 y - 2 |$
$y = | - 3 y + 18 |$
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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