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Solution - Absolute value equations

Exact form:

= \frac{3}{2}, \frac{9}{2}

=\frac{3}{2} , \frac{9}{2}

Mixed number form:

= 1 \frac{1}{2}, 4 \frac{1}{2}

=1\frac{1}{2} , 4\frac{1}{2}

Decimal form:

= 1.5, 4.5

=1.5 , 4.5

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

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

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:

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

2. Solve the two equations for

7 additional steps

$- 3 = 2 \cdot (z - 3)$

Expand the parentheses:

$- 3 = 2 z + 2 \cdot - 3$

Simplify the arithmetic:

$- 3 = 2 z - 6$

Swap sides:

$2 z - 6 = - 3$

Add to both sides:

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

Simplify the arithmetic:

$2 z = - 3 + 6$

Simplify the arithmetic:

$2 z = 3$

Divide both sides by :

$\frac{(2 z)}{2} = \frac{3}{2}$

Simplify the fraction:

$z = \frac{3}{2}$

12 additional steps

$- 3 = 2 \cdot (- (z - 3))$

Expand the parentheses:

$- 3 = 2 \cdot (- z + 3)$

$- 3 = 2 \cdot - z + 2 \cdot 3$

Group like terms:

$- 3 = (2 \cdot - 1) z + 2 \cdot 3$

Multiply the coefficients:

$- 3 = - 2 z + 2 \cdot 3$

Simplify the arithmetic:

$- 3 = - 2 z + 6$

Swap sides:

$- 2 z + 6 = - 3$

Subtract from both sides:

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

Simplify the arithmetic:

$- 2 z = - 3 - 6$

Simplify the arithmetic:

$- 2 z = - 9$

Divide both sides by :

$\frac{(- 2 z)}{- 2} = \frac{- 9}{- 2}$

Cancel out the negatives:

$\frac{2 z}{2} = \frac{- 9}{- 2}$

Simplify the fraction:

$z = \frac{- 9}{- 2}$

Cancel out the negatives:

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

3. List the solutions

$= \frac{3}{2}, \frac{9}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 3 |$
$y = 2 | z - 3 |$
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 without absolute value bars

2. Solve the two equations for

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

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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