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Solution - Absolute value equations

Exact form:

z = 10, - \frac{2}{7}

z=10 , -\frac{2}{7}

Decimal form:

z = 10, - 0.286

z=10 , -0.286

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
$| 4 z - 4 | = | 3 z + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 z - 4 \| = \| 3 z + 6 \|$
$x = + y$	$(4 z - 4) = (3 z + 6)$
$x = - y$	$(4 z - 4) = - (3 z + 6)$
$+ x = y$	$(4 z - 4) = (3 z + 6)$
$- x = y$	$- (4 z - 4) = (3 z + 6)$

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:

$\| x \| = \| y \|$	$\| 4 z - 4 \| = \| 3 z + 6 \|$
$x = + y$ , $+ x = y$	$(4 z - 4) = (3 z + 6)$
$x = - y$ , $- x = y$	$(4 z - 4) = - (3 z + 6)$

2. Solve the two equations for z

7 additional steps

$(4 z - 4) = (3 z + 6)$

Subtract from both sides:

$(4 z - 4) - 3 z = (3 z + 6) - 3 z$

Group like terms:

$(4 z - 3 z) - 4 = (3 z + 6) - 3 z$

Simplify the arithmetic:

$z - 4 = (3 z + 6) - 3 z$

Group like terms:

$z - 4 = (3 z - 3 z) + 6$

Simplify the arithmetic:

$z - 4 = 6$

Add to both sides:

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

Simplify the arithmetic:

$z = 6 + 4$

Simplify the arithmetic:

$z = 10$

10 additional steps

$(4 z - 4) = - (3 z + 6)$

Expand the parentheses:

$(4 z - 4) = - 3 z - 6$

Add to both sides:

$(4 z - 4) + 3 z = (- 3 z - 6) + 3 z$

Group like terms:

$(4 z + 3 z) - 4 = (- 3 z - 6) + 3 z$

Simplify the arithmetic:

$7 z - 4 = (- 3 z - 6) + 3 z$

Group like terms:

$7 z - 4 = (- 3 z + 3 z) - 6$

Simplify the arithmetic:

$7 z - 4 = - 6$

Add to both sides:

$(7 z - 4) + 4 = - 6 + 4$

Simplify the arithmetic:

$7 z = - 6 + 4$

Simplify the arithmetic:

$7 z = - 2$

Divide both sides by :

$\frac{(7 z)}{7} = \frac{- 2}{7}$

Simplify the fraction:

$z = \frac{- 2}{7}$

3. List the solutions

$z = 10, - \frac{2}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 z - 4 |$
$y = | 3 z + 6 |$
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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