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Solution - Absolute value equations

Exact form:

y = - 3

y=-3

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

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

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:

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

2. Solve the two equations for y

5 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 4 = 10$

The statement is false:

$- 4 = 10$

The equation is false so it has no solution.

12 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 y - 4 = - 10$

Add to 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:

$y = - 3$

3. Graph

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