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Solution - Absolute value equations

Exact form:

y = - 2, \frac{2}{7}

y=-2 , \frac{2}{7}

Decimal form:

y = - 2, 0.286

y=-2 , 0.286

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
$| 5 y - 6 | = | 9 y + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 y - 6 \| = \| 9 y + 2 \|$
$x = + y$	$(5 y - 6) = (9 y + 2)$
$x = - y$	$(5 y - 6) = - (9 y + 2)$
$+ x = y$	$(5 y - 6) = (9 y + 2)$
$- x = y$	$- (5 y - 6) = (9 y + 2)$

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:

$\| x \| = \| y \|$	$\| 5 y - 6 \| = \| 9 y + 2 \|$
$x = + y$ , $+ x = y$	$(5 y - 6) = (9 y + 2)$
$x = - y$ , $- x = y$	$(5 y - 6) = - (9 y + 2)$

2. Solve the two equations for y

13 additional steps

$(5 y - 6) = (9 y + 2)$

Subtract from both sides:

$(5 y - 6) - 9 y = (9 y + 2) - 9 y$

Group like terms:

$(5 y - 9 y) - 6 = (9 y + 2) - 9 y$

Simplify the arithmetic:

$- 4 y - 6 = (9 y + 2) - 9 y$

Group like terms:

$- 4 y - 6 = (9 y - 9 y) + 2$

Simplify the arithmetic:

$- 4 y - 6 = 2$

Add to both sides:

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

Simplify the arithmetic:

$- 4 y = 2 + 6$

Simplify the arithmetic:

$- 4 y = 8$

Divide both sides by :

$\frac{(- 4 y)}{- 4} = \frac{8}{- 4}$

Cancel out the negatives:

$\frac{4 y}{4} = \frac{8}{- 4}$

Simplify the fraction:

$y = \frac{8}{- 4}$

Move the negative sign from the denominator to the numerator:

$y = \frac{- 8}{4}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(- 2 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$y = - 2$

12 additional steps

$(5 y - 6) = - (9 y + 2)$

Expand the parentheses:

$(5 y - 6) = - 9 y - 2$

Add to both sides:

$(5 y - 6) + 9 y = (- 9 y - 2) + 9 y$

Group like terms:

$(5 y + 9 y) - 6 = (- 9 y - 2) + 9 y$

Simplify the arithmetic:

$14 y - 6 = (- 9 y - 2) + 9 y$

Group like terms:

$14 y - 6 = (- 9 y + 9 y) - 2$

Simplify the arithmetic:

$14 y - 6 = - 2$

Add to both sides:

$(14 y - 6) + 6 = - 2 + 6$

Simplify the arithmetic:

$14 y = - 2 + 6$

Simplify the arithmetic:

$14 y = 4$

Divide both sides by :

$\frac{(14 y)}{14} = \frac{4}{14}$

Simplify the fraction:

$y = \frac{4}{14}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$y = \frac{2}{7}$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 y - 6 |$
$y = | 9 y + 2 |$
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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