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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

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

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

Decimal form:

z = 1.5, - 1

z=1.5 , -1

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
$| z + 6 | = 5 | z |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for z

12 additional steps

$(z + 6) = 5 z$

Subtract from both sides:

$(z + 6) - 5 z = (5 z) - 5 z$

Group like terms:

$(z - 5 z) + 6 = (5 z) - 5 z$

Simplify the arithmetic:

$- 4 z + 6 = (5 z) - 5 z$

Simplify the arithmetic:

$- 4 z + 6 = 0$

Subtract from both sides:

$(- 4 z + 6) - 6 = 0 - 6$

Simplify the arithmetic:

$- 4 z = 0 - 6$

Simplify the arithmetic:

$- 4 z = - 6$

Divide both sides by :

$\frac{(- 4 z)}{- 4} = \frac{- 6}{- 4}$

Cancel out the negatives:

$\frac{4 z}{4} = \frac{- 6}{- 4}$

Simplify the fraction:

$z = \frac{- 6}{- 4}$

Cancel out the negatives:

$z = \frac{6}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

11 additional steps

$(z + 6) = 5 \cdot - z$

Group like terms:

$(z + 6) = (5 \cdot - 1) z$

Multiply the coefficients:

$(z + 6) = - 5 z$

Add to both sides:

$(z + 6) + 5 z = (- 5 z) + 5 z$

Group like terms:

$(z + 5 z) + 6 = (- 5 z) + 5 z$

Simplify the arithmetic:

$6 z + 6 = (- 5 z) + 5 z$

Simplify the arithmetic:

$6 z + 6 = 0$

Subtract from both sides:

$(6 z + 6) - 6 = 0 - 6$

Simplify the arithmetic:

$6 z = 0 - 6$

Simplify the arithmetic:

$6 z = - 6$

Divide both sides by :

$\frac{(6 z)}{6} = \frac{- 6}{6}$

Simplify the fraction:

$z = \frac{- 6}{6}$

Simplify the fraction:

$z = - 1$

3. List the solutions

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

4. Graph

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

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 z

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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