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Solution - Absolute value equations

Exact form:

y = - 3, 1

y=-3 , 1

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| y - 3 | - | 2 y | = 0$

Add $| 2 y |$ to both sides of the equation:

$| y - 3 | - | 2 y | + | 2 y | = | 2 y |$

Simplify the arithmetic

$| y - 3 | = | 2 y |$

2. 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
$| y - 3 | = | 2 y |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| y - 3 \| = \| 2 y \|$
$x = + y$	$(y - 3) = (2 y)$
$x = - y$	$(y - 3) = (- (2 y))$
$+ x = y$	$(y - 3) = (2 y)$
$- x = y$	$- (y - 3) = (2 y)$

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:

$\| x \| = \| y \|$	$\| y - 3 \| = \| 2 y \|$
$x = + y$ , $+ x = y$	$(y - 3) = (2 y)$
$x = - y$ , $- x = y$	$(y - 3) = (- (2 y))$

3. Solve the two equations for y

9 additional steps

$(y - 3) = 2 y$

Subtract from both sides:

$(y - 3) - 2 y = (2 y) - 2 y$

Group like terms:

$(y - 2 y) - 3 = (2 y) - 2 y$

Simplify the arithmetic:

$- y - 3 = (2 y) - 2 y$

Simplify the arithmetic:

$- y - 3 = 0$

Add to both sides:

$(- y - 3) + 3 = 0 + 3$

Simplify the arithmetic:

$- y = 0 + 3$

Simplify the arithmetic:

$- y = 3$

Multiply both sides by :

$- y \cdot - 1 = 3 \cdot - 1$

Remove the one(s):

$y = 3 \cdot - 1$

Simplify the arithmetic:

$y = - 3$

8 additional steps

$(y - 3) = - 2 y$

Add to both sides:

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

Simplify the arithmetic:

$y = (- 2 y) + 3$

Add to both sides:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$3 y = 3$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$y = 1$

4. List the solutions

$y = - 3, 1$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | y - 3 |$
$y = | 2 y |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for y

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for y

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved