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Solution - Absolute value equations

Exact form:

y = 16, \frac{2}{9}

y=16 , \frac{2}{9}

Decimal form:

y = 16, 0.222

y=16 , 0.222

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
$| 5 y - 9 | = | 4 y + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 y - 9 \| = \| 4 y + 7 \|$
$x = + y$	$(5 y - 9) = (4 y + 7)$
$x = - y$	$(5 y - 9) = - (4 y + 7)$
$+ x = y$	$(5 y - 9) = (4 y + 7)$
$- x = y$	$- (5 y - 9) = (4 y + 7)$

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:

$\| x \| = \| y \|$	$\| 5 y - 9 \| = \| 4 y + 7 \|$
$x = + y$ , $+ x = y$	$(5 y - 9) = (4 y + 7)$
$x = - y$ , $- x = y$	$(5 y - 9) = - (4 y + 7)$

2. Solve the two equations for y

7 additional steps

$(5 y - 9) = (4 y + 7)$

Subtract from both sides:

$(5 y - 9) - 4 y = (4 y + 7) - 4 y$

Group like terms:

$(5 y - 4 y) - 9 = (4 y + 7) - 4 y$

Simplify the arithmetic:

$y - 9 = (4 y + 7) - 4 y$

Group like terms:

$y - 9 = (4 y - 4 y) + 7$

Simplify the arithmetic:

$y - 9 = 7$

Add to both sides:

$(y - 9) + 9 = 7 + 9$

Simplify the arithmetic:

$y = 7 + 9$

Simplify the arithmetic:

$y = 16$

10 additional steps

$(5 y - 9) = - (4 y + 7)$

Expand the parentheses:

$(5 y - 9) = - 4 y - 7$

Add to both sides:

$(5 y - 9) + 4 y = (- 4 y - 7) + 4 y$

Group like terms:

$(5 y + 4 y) - 9 = (- 4 y - 7) + 4 y$

Simplify the arithmetic:

$9 y - 9 = (- 4 y - 7) + 4 y$

Group like terms:

$9 y - 9 = (- 4 y + 4 y) - 7$

Simplify the arithmetic:

$9 y - 9 = - 7$

Add to both sides:

$(9 y - 9) + 9 = - 7 + 9$

Simplify the arithmetic:

$9 y = - 7 + 9$

Simplify the arithmetic:

$9 y = 2$

Divide both sides by :

$\frac{(9 y)}{9} = \frac{2}{9}$

Simplify the fraction:

$y = \frac{2}{9}$

3. List the solutions

$y = 16, \frac{2}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 y - 9 |$
$y = | 4 y + 7 |$
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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