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Solution - Absolute value equations

Exact form:

y = - \frac{4}{3}, - \frac{2}{5}

y=-\frac{4}{3} , -\frac{2}{5}

Mixed number form:

y = - 1 \frac{1}{3}, - \frac{2}{5}

y=-1\frac{1}{3} , -\frac{2}{5}

Decimal form:

y = - 1.333, - 0.4

y=-1.333 , -0.4

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
$| - 4 y - 3 | = | - y + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 4 y - 3 \| = \| - y + 1 \|$
$x = + y$	$(- 4 y - 3) = (- y + 1)$
$x = - y$	$(- 4 y - 3) = - (- y + 1)$
$+ x = y$	$(- 4 y - 3) = (- y + 1)$
$- x = y$	$- (- 4 y - 3) = (- y + 1)$

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:

$\| x \| = \| y \|$	$\| - 4 y - 3 \| = \| - y + 1 \|$
$x = + y$ , $+ x = y$	$(- 4 y - 3) = (- y + 1)$
$x = - y$ , $- x = y$	$(- 4 y - 3) = - (- y + 1)$

2. Solve the two equations for y

11 additional steps

$(- 4 y - 3) = (- y + 1)$

Add to both sides:

$(- 4 y - 3) + y = (- y + 1) + y$

Group like terms:

$(- 4 y + y) - 3 = (- y + 1) + y$

Simplify the arithmetic:

$- 3 y - 3 = (- y + 1) + y$

Group like terms:

$- 3 y - 3 = (- y + y) + 1$

Simplify the arithmetic:

$- 3 y - 3 = 1$

Add to both sides:

$(- 3 y - 3) + 3 = 1 + 3$

Simplify the arithmetic:

$- 3 y = 1 + 3$

Simplify the arithmetic:

$- 3 y = 4$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

12 additional steps

$(- 4 y - 3) = - (- y + 1)$

Expand the parentheses:

$(- 4 y - 3) = y - 1$

Subtract from both sides:

$(- 4 y - 3) - y = (y - 1) - y$

Group like terms:

$(- 4 y - y) - 3 = (y - 1) - y$

Simplify the arithmetic:

$- 5 y - 3 = (y - 1) - y$

Group like terms:

$- 5 y - 3 = (y - y) - 1$

Simplify the arithmetic:

$- 5 y - 3 = - 1$

Add to both sides:

$(- 5 y - 3) + 3 = - 1 + 3$

Simplify the arithmetic:

$- 5 y = - 1 + 3$

Simplify the arithmetic:

$- 5 y = 2$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

3. List the solutions

$y = - \frac{4}{3}, - \frac{2}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 4 y - 3 |$
$y = | - y + 1 |$
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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