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Solution - Absolute value equations

Exact form:

y = - \frac{1}{2}

y=-\frac{1}{2}

Decimal form:

y = - 0.5

y=-0.5

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

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

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:

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

2. Solve the two equations for y

11 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$10 y - 4 = (- 5 y - 9) + 5 y$

Group like terms:

$10 y - 4 = (- 5 y + 5 y) - 9$

Simplify the arithmetic:

$10 y - 4 = - 9$

Add to both sides:

$(10 y - 4) + 4 = - 9 + 4$

Simplify the arithmetic:

$10 y = - 9 + 4$

Simplify the arithmetic:

$10 y = - 5$

Divide both sides by :

$\frac{(10 y)}{10} = \frac{- 5}{10}$

Simplify the fraction:

$y = \frac{- 5}{10}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(- 1 \cdot 5)}{(2 \cdot 5)}$

Factor out and cancel the greatest common factor:

$y = \frac{- 1}{2}$

6 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 4 = 9$

The statement is false:

$- 4 = 9$

The equation is false so it has no solution.

3. List the solutions

$y = - \frac{1}{2}$
(1 solution(s))

4. Graph

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