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Solution - Absolute value equations

Exact form:

z = - 1, 1

z=-1 , 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
$| - 6 z + 7 | = | - 7 z + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 6 z + 7 \| = \| - 7 z + 6 \|$
$x = + y$	$(- 6 z + 7) = (- 7 z + 6)$
$x = - y$	$(- 6 z + 7) = - (- 7 z + 6)$
$+ x = y$	$(- 6 z + 7) = (- 7 z + 6)$
$- x = y$	$- (- 6 z + 7) = (- 7 z + 6)$

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:

$\| x \| = \| y \|$	$\| - 6 z + 7 \| = \| - 7 z + 6 \|$
$x = + y$ , $+ x = y$	$(- 6 z + 7) = (- 7 z + 6)$
$x = - y$ , $- x = y$	$(- 6 z + 7) = - (- 7 z + 6)$

2. Solve the two equations for z

7 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$z + 7 = 6$

Subtract from both sides:

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

Simplify the arithmetic:

$z = 6 - 7$

Simplify the arithmetic:

$z = - 1$

13 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 13 z + 7 = (7 z - 6) - 7 z$

Group like terms:

$- 13 z + 7 = (7 z - 7 z) - 6$

Simplify the arithmetic:

$- 13 z + 7 = - 6$

Subtract from both sides:

$(- 13 z + 7) - 7 = - 6 - 7$

Simplify the arithmetic:

$- 13 z = - 6 - 7$

Simplify the arithmetic:

$- 13 z = - 13$

Divide both sides by :

$\frac{(- 13 z)}{- 13} = \frac{- 13}{- 13}$

Cancel out the negatives:

$\frac{13 z}{13} = \frac{- 13}{- 13}$

Simplify the fraction:

$z = \frac{- 13}{- 13}$

Cancel out the negatives:

$z = \frac{13}{13}$

Simplify the fraction:

$z = 1$

3. List the solutions

$z = - 1, 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 6 z + 7 |$
$y = | - 7 z + 6 |$
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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