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Solution - Absolute value equations

Exact form:

y = 4, 40

y=4 , 40

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
$| 5 y - 2 | = | - 6 y + 42 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 y - 2 \| = \| - 6 y + 42 \|$
$x = + y$	$(5 y - 2) = (- 6 y + 42)$
$x = - y$	$(5 y - 2) = - (- 6 y + 42)$
$+ x = y$	$(5 y - 2) = (- 6 y + 42)$
$- x = y$	$- (5 y - 2) = (- 6 y + 42)$

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:

$\| x \| = \| y \|$	$\| 5 y - 2 \| = \| - 6 y + 42 \|$
$x = + y$ , $+ x = y$	$(5 y - 2) = (- 6 y + 42)$
$x = - y$ , $- x = y$	$(5 y - 2) = - (- 6 y + 42)$

2. Solve the two equations for y

11 additional steps

$(5 y - 2) = (- 6 y + 42)$

Add to both sides:

$(5 y - 2) + 6 y = (- 6 y + 42) + 6 y$

Group like terms:

$(5 y + 6 y) - 2 = (- 6 y + 42) + 6 y$

Simplify the arithmetic:

$11 y - 2 = (- 6 y + 42) + 6 y$

Group like terms:

$11 y - 2 = (- 6 y + 6 y) + 42$

Simplify the arithmetic:

$11 y - 2 = 42$

Add to both sides:

$(11 y - 2) + 2 = 42 + 2$

Simplify the arithmetic:

$11 y = 42 + 2$

Simplify the arithmetic:

$11 y = 44$

Divide both sides by :

$\frac{(11 y)}{11} = \frac{44}{11}$

Simplify the fraction:

$y = \frac{44}{11}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(4 \cdot 11)}{(1 \cdot 11)}$

Factor out and cancel the greatest common factor:

$y = 4$

11 additional steps

$(5 y - 2) = - (- 6 y + 42)$

Expand the parentheses:

$(5 y - 2) = 6 y - 42$

Subtract from both sides:

$(5 y - 2) - 6 y = (6 y - 42) - 6 y$

Group like terms:

$(5 y - 6 y) - 2 = (6 y - 42) - 6 y$

Simplify the arithmetic:

$- y - 2 = (6 y - 42) - 6 y$

Group like terms:

$- y - 2 = (6 y - 6 y) - 42$

Simplify the arithmetic:

$- y - 2 = - 42$

Add to both sides:

$(- y - 2) + 2 = - 42 + 2$

Simplify the arithmetic:

$- y = - 42 + 2$

Simplify the arithmetic:

$- y = - 40$

Multiply both sides by :

$- y \cdot - 1 = - 40 \cdot - 1$

Remove the one(s):

$y = - 40 \cdot - 1$

Simplify the arithmetic:

$y = 40$

3. List the solutions

$y = 4, 40$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 y - 2 |$
$y = | - 6 y + 42 |$
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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