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Solution - Absolute value equations

Exact form:

y = - \frac{4}{3}

y=-\frac{4}{3}

Mixed number form:

y = - 1 \frac{1}{3}

y=-1\frac{1}{3}

Decimal form:

y = - 1.333

y=-1.333

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
$| 6 y - 2 | = | 6 y + 18 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 y - 2 \| = \| 6 y + 18 \|$
$x = + y$	$(6 y - 2) = (6 y + 18)$
$x = - y$	$(6 y - 2) = - (6 y + 18)$
$+ x = y$	$(6 y - 2) = (6 y + 18)$
$- x = y$	$- (6 y - 2) = (6 y + 18)$

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:

$\| x \| = \| y \|$	$\| 6 y - 2 \| = \| 6 y + 18 \|$
$x = + y$ , $+ x = y$	$(6 y - 2) = (6 y + 18)$
$x = - y$ , $- x = y$	$(6 y - 2) = - (6 y + 18)$

2. Solve the two equations for y

5 additional steps

$(6 y - 2) = (6 y + 18)$

Subtract from both sides:

$(6 y - 2) - 6 y = (6 y + 18) - 6 y$

Group like terms:

$(6 y - 6 y) - 2 = (6 y + 18) - 6 y$

Simplify the arithmetic:

$- 2 = (6 y + 18) - 6 y$

Group like terms:

$- 2 = (6 y - 6 y) + 18$

Simplify the arithmetic:

$- 2 = 18$

The statement is false:

$- 2 = 18$

The equation is false so it has no solution.

12 additional steps

$(6 y - 2) = - (6 y + 18)$

Expand the parentheses:

$(6 y - 2) = - 6 y - 18$

Add to both sides:

$(6 y - 2) + 6 y = (- 6 y - 18) + 6 y$

Group like terms:

$(6 y + 6 y) - 2 = (- 6 y - 18) + 6 y$

Simplify the arithmetic:

$12 y - 2 = (- 6 y - 18) + 6 y$

Group like terms:

$12 y - 2 = (- 6 y + 6 y) - 18$

Simplify the arithmetic:

$12 y - 2 = - 18$

Add to both sides:

$(12 y - 2) + 2 = - 18 + 2$

Simplify the arithmetic:

$12 y = - 18 + 2$

Simplify the arithmetic:

$12 y = - 16$

Divide both sides by :

$\frac{(12 y)}{12} = \frac{- 16}{12}$

Simplify the fraction:

$y = \frac{- 16}{12}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(- 4 \cdot 4)}{(3 \cdot 4)}$

Factor out and cancel the greatest common factor:

$y = \frac{- 4}{3}$

3. Graph

Each line represents the function of one side of the equation:
$y = | 6 y - 2 |$
$y = | 6 y + 18 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved