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Solution - Absolute value equations

Exact form:

z = 0

z=0

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
$| z + 17 | = | z - 17 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| z + 17 \| = \| z - 17 \|$
$x = + y$	$(z + 17) = (z - 17)$
$x = - y$	$(z + 17) = - (z - 17)$
$+ x = y$	$(z + 17) = (z - 17)$
$- x = y$	$- (z + 17) = (z - 17)$

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:

$\| x \| = \| y \|$	$\| z + 17 \| = \| z - 17 \|$
$x = + y$ , $+ x = y$	$(z + 17) = (z - 17)$
$x = - y$ , $- x = y$	$(z + 17) = - (z - 17)$

2. Solve the two equations for z

5 additional steps

$(z + 17) = (z - 17)$

Subtract from both sides:

$(z + 17) - z = (z - 17) - z$

Group like terms:

$(z - z) + 17 = (z - 17) - z$

Simplify the arithmetic:

$17 = (z - 17) - z$

Group like terms:

$17 = (z - z) - 17$

Simplify the arithmetic:

$17 = - 17$

The statement is false:

$17 = - 17$

The equation is false so it has no solution.

9 additional steps

$(z + 17) = - (z - 17)$

Expand the parentheses:

$(z + 17) = - z + 17$

Add to both sides:

$(z + 17) + z = (- z + 17) + z$

Group like terms:

$(z + z) + 17 = (- z + 17) + z$

Simplify the arithmetic:

$2 z + 17 = (- z + 17) + z$

Group like terms:

$2 z + 17 = (- z + z) + 17$

Simplify the arithmetic:

$2 z + 17 = 17$

Subtract from both sides:

$(2 z + 17) - 17 = 17 - 17$

Simplify the arithmetic:

$2 z = 17 - 17$

Simplify the arithmetic:

$2 z = 0$

Divide both sides by the coefficient:

$z = 0$

3. Graph

Each line represents the function of one side of the equation:
$y = | z + 17 |$
$y = | z - 17 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved