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Solution - Absolute value equations

Exact form:

y = \frac{1}{2}, - \frac{3}{2}

y=\frac{1}{2} , -\frac{3}{2}

Mixed number form:

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

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

Decimal form:

y = 0.5, - 1.5

y=0.5 , -1.5

See steps

Step-by-step explanation

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

$| 4 y | + | 2 y - 3 | = 0$

Add $- | 2 y - 3 |$ to both sides of the equation:

$| 4 y | + | 2 y - 3 | - | 2 y - 3 | = - | 2 y - 3 |$

Simplify the arithmetic

$| 4 y | = - | 2 y - 3 |$

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
$| 4 y | = - | 2 y - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 y \| = - \| 2 y - 3 \|$
$x = + y$	$(4 y) = - (2 y - 3)$
$x = - y$	$(4 y) = - - (2 y - 3)$
$+ x = y$	$(4 y) = - (2 y - 3)$
$- x = y$	$- (4 y) = - (2 y - 3)$

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:

$\| x \| = \| y \|$	$\| 4 y \| = - \| 2 y - 3 \|$
$x = + y$ , $+ x = y$	$(4 y) = - (2 y - 3)$
$x = - y$ , $- x = y$	$(4 y) = - - (2 y - 3)$

3. Solve the two equations for y

8 additional steps

$4 y = - (2 y - 3)$

Expand the parentheses:

$4 y = - 2 y + 3$

Add to both sides:

$(4 y) + 2 y = (- 2 y + 3) + 2 y$

Simplify the arithmetic:

$6 y = (- 2 y + 3) + 2 y$

Group like terms:

$6 y = (- 2 y + 2 y) + 3$

Simplify the arithmetic:

$6 y = 3$

Divide both sides by :

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

Simplify the fraction:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

6 additional steps

$4 y = - (- (2 y - 3))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$4 y = 2 y - 3$

Subtract from both sides:

$(4 y) - 2 y = (2 y - 3) - 2 y$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 y = - 3$

Divide both sides by :

$\frac{(2 y)}{2} = \frac{- 3}{2}$

Simplify the fraction:

$y = \frac{- 3}{2}$

4. List the solutions

$y = \frac{1}{2}, - \frac{3}{2}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 4 y |$
$y = - | 2 y - 3 |$
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