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Solution - Absolute value equations

Exact form:

y = - \frac{7}{4}

y=-\frac{7}{4}

Mixed number form:

y = - 1 \frac{3}{4}

y=-1\frac{3}{4}

Decimal form:

y = - 1.75

y=-1.75

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
$| - 4 y - 5 | = | 4 y + 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 4 y - 5 \| = \| 4 y + 9 \|$
$x = + y$	$(- 4 y - 5) = (4 y + 9)$
$x = - y$	$(- 4 y - 5) = - (4 y + 9)$
$+ x = y$	$(- 4 y - 5) = (4 y + 9)$
$- x = y$	$- (- 4 y - 5) = (4 y + 9)$

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:

$\| x \| = \| y \|$	$\| - 4 y - 5 \| = \| 4 y + 9 \|$
$x = + y$ , $+ x = y$	$(- 4 y - 5) = (4 y + 9)$
$x = - y$ , $- x = y$	$(- 4 y - 5) = - (4 y + 9)$

2. Solve the two equations for y

13 additional steps

$(- 4 y - 5) = (4 y + 9)$

Subtract from both sides:

$(- 4 y - 5) - 4 y = (4 y + 9) - 4 y$

Group like terms:

$(- 4 y - 4 y) - 5 = (4 y + 9) - 4 y$

Simplify the arithmetic:

$- 8 y - 5 = (4 y + 9) - 4 y$

Group like terms:

$- 8 y - 5 = (4 y - 4 y) + 9$

Simplify the arithmetic:

$- 8 y - 5 = 9$

Add to both sides:

$(- 8 y - 5) + 5 = 9 + 5$

Simplify the arithmetic:

$- 8 y = 9 + 5$

Simplify the arithmetic:

$- 8 y = 14$

Divide both sides by :

$\frac{(- 8 y)}{- 8} = \frac{14}{- 8}$

Cancel out the negatives:

$\frac{8 y}{8} = \frac{14}{- 8}$

Simplify the fraction:

$y = \frac{14}{- 8}$

Move the negative sign from the denominator to the numerator:

$y = \frac{- 14}{8}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(- 7 \cdot 2)}{(4 \cdot 2)}$

Factor out and cancel the greatest common factor:

$y = \frac{- 7}{4}$

6 additional steps

$(- 4 y - 5) = - (4 y + 9)$

Expand the parentheses:

$(- 4 y - 5) = - 4 y - 9$

Add to both sides:

$(- 4 y - 5) + 4 y = (- 4 y - 9) + 4 y$

Group like terms:

$(- 4 y + 4 y) - 5 = (- 4 y - 9) + 4 y$

Simplify the arithmetic:

$- 5 = (- 4 y - 9) + 4 y$

Group like terms:

$- 5 = (- 4 y + 4 y) - 9$

Simplify the arithmetic:

$- 5 = - 9$

The statement is false:

$- 5 = - 9$

The equation is false so it has no solution.

3. List the solutions

$y = - \frac{7}{4}$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 4 y - 5 |$
$y = | 4 y + 9 |$
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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