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Solution - Absolute value equations

Exact form:

y = - 4, 2

y=-4 , 2

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
$| y - 5 | = | 2 y - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| y - 5 \| = \| 2 y - 1 \|$
$x = + y$	$(y - 5) = (2 y - 1)$
$x = - y$	$(y - 5) = - (2 y - 1)$
$+ x = y$	$(y - 5) = (2 y - 1)$
$- x = y$	$- (y - 5) = (2 y - 1)$

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:

$\| x \| = \| y \|$	$\| y - 5 \| = \| 2 y - 1 \|$
$x = + y$ , $+ x = y$	$(y - 5) = (2 y - 1)$
$x = - y$ , $- x = y$	$(y - 5) = - (2 y - 1)$

2. Solve the two equations for y

10 additional steps

$(y - 5) = (2 y - 1)$

Subtract from both sides:

$(y - 5) - 2 y = (2 y - 1) - 2 y$

Group like terms:

$(y - 2 y) - 5 = (2 y - 1) - 2 y$

Simplify the arithmetic:

$- y - 5 = (2 y - 1) - 2 y$

Group like terms:

$- y - 5 = (2 y - 2 y) - 1$

Simplify the arithmetic:

$- y - 5 = - 1$

Add to both sides:

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

Simplify the arithmetic:

$- y = - 1 + 5$

Simplify the arithmetic:

$- y = 4$

Multiply both sides by :

$- y \cdot - 1 = 4 \cdot - 1$

Remove the one(s):

$y = 4 \cdot - 1$

Simplify the arithmetic:

$y = - 4$

12 additional steps

$(y - 5) = - (2 y - 1)$

Expand the parentheses:

$(y - 5) = - 2 y + 1$

Add to both sides:

$(y - 5) + 2 y = (- 2 y + 1) + 2 y$

Group like terms:

$(y + 2 y) - 5 = (- 2 y + 1) + 2 y$

Simplify the arithmetic:

$3 y - 5 = (- 2 y + 1) + 2 y$

Group like terms:

$3 y - 5 = (- 2 y + 2 y) + 1$

Simplify the arithmetic:

$3 y - 5 = 1$

Add to both sides:

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

Simplify the arithmetic:

$3 y = 1 + 5$

Simplify the arithmetic:

$3 y = 6$

Divide both sides by :

$\frac{(3 y)}{3} = \frac{6}{3}$

Simplify the fraction:

$y = \frac{6}{3}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(2 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$y = 2$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | y - 5 |$
$y = | 2 y - 1 |$
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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