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Solution - Absolute value equations

Exact form:

z = 19, - \frac{1}{13}

z=19 , -\frac{1}{13}

Decimal form:

z = 19, - 0.077

z=19 , -0.077

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
$| 7 z - 9 | = | 6 z + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 7 z - 9 \| = \| 6 z + 10 \|$
$x = + y$	$(7 z - 9) = (6 z + 10)$
$x = - y$	$(7 z - 9) = - (6 z + 10)$
$+ x = y$	$(7 z - 9) = (6 z + 10)$
$- x = y$	$- (7 z - 9) = (6 z + 10)$

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:

$\| x \| = \| y \|$	$\| 7 z - 9 \| = \| 6 z + 10 \|$
$x = + y$ , $+ x = y$	$(7 z - 9) = (6 z + 10)$
$x = - y$ , $- x = y$	$(7 z - 9) = - (6 z + 10)$

2. Solve the two equations for z

7 additional steps

$(7 z - 9) = (6 z + 10)$

Subtract from both sides:

$(7 z - 9) - 6 z = (6 z + 10) - 6 z$

Group like terms:

$(7 z - 6 z) - 9 = (6 z + 10) - 6 z$

Simplify the arithmetic:

$z - 9 = (6 z + 10) - 6 z$

Group like terms:

$z - 9 = (6 z - 6 z) + 10$

Simplify the arithmetic:

$z - 9 = 10$

Add to both sides:

$(z - 9) + 9 = 10 + 9$

Simplify the arithmetic:

$z = 10 + 9$

Simplify the arithmetic:

$z = 19$

10 additional steps

$(7 z - 9) = - (6 z + 10)$

Expand the parentheses:

$(7 z - 9) = - 6 z - 10$

Add to both sides:

$(7 z - 9) + 6 z = (- 6 z - 10) + 6 z$

Group like terms:

$(7 z + 6 z) - 9 = (- 6 z - 10) + 6 z$

Simplify the arithmetic:

$13 z - 9 = (- 6 z - 10) + 6 z$

Group like terms:

$13 z - 9 = (- 6 z + 6 z) - 10$

Simplify the arithmetic:

$13 z - 9 = - 10$

Add to both sides:

$(13 z - 9) + 9 = - 10 + 9$

Simplify the arithmetic:

$13 z = - 10 + 9$

Simplify the arithmetic:

$13 z = - 1$

Divide both sides by :

$\frac{(13 z)}{13} = \frac{- 1}{13}$

Simplify the fraction:

$z = \frac{- 1}{13}$

3. List the solutions

$z = 19, - \frac{1}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 7 z - 9 |$
$y = | 6 z + 10 |$
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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