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Penyelesaian - Absolute value equations

Hasil Akhir Langkah-langkah Terma dan topik

Exact form:

z = 0

z=0

Lihat langkah Learn more with Tiger Try this:

Penjelasan langkah demi langkah

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

$\| x \| = \| y \|$	$\| z - 1 \| = \| z + 1 \|$
$x = + y$	$(z - 1) = (z + 1)$
$x = - y$	$(z - 1) = - (z + 1)$
$+ x = y$	$(z - 1) = (z + 1)$
$- x = y$	$- (z - 1) = (z + 1)$

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:

$\| x \| = \| y \|$	$\| z - 1 \| = \| z + 1 \|$
$x = + y$ , $+ x = y$	$(z - 1) = (z + 1)$
$x = - y$ , $- x = y$	$(z - 1) = - (z + 1)$

2. Solve the two equations for z

5 additional steps

$(z - 1) = (z + 1)$

Subtract from both sides:

$(z - 1) - z = (z + 1) - z$

Kumpulkan sebutan sejenis:

$(z - z) - 1 = (z + 1) - z$

Permudahkan aritmetik:

$- 1 = (z + 1) - z$

Kumpulkan sebutan sejenis:

$- 1 = (z - z) + 1$

Permudahkan aritmetik:

$- 1 = 1$

Pernyataan ini palsu:

$- 1 = 1$

The equation is false so it has no solution.

9 additional steps

$(z - 1) = - (z + 1)$

Expand the parentheses:

$(z - 1) = - z - 1$

Add to both sides:

$(z - 1) + z = (- z - 1) + z$

Kumpulkan sebutan sejenis:

$(z + z) - 1 = (- z - 1) + z$

Permudahkan aritmetik:

$2 z - 1 = (- z - 1) + z$

Kumpulkan sebutan sejenis:

$2 z - 1 = (- z + z) - 1$

Permudahkan aritmetik:

$2 z - 1 = - 1$

Add to both sides:

$(2 z - 1) + 1 = - 1 + 1$

Permudahkan aritmetik:

$2 z = - 1 + 1$

Permudahkan aritmetik:

$2 z = 0$

Divide both sides by the coefficient:

$z = 0$

3. Graph

Each line represents the function of one side of the equation:
$y = | z - 1 |$
$y = | z + 1 |$
The equation is true where the two lines cross.

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Mengapa belajar ini

Learn more with Tiger

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.

Penyelesaian - Absolute value equations

Penjelasan langkah demi langkah

1. Rewrite the equation without absolute value bars

2. Solve the two equations for z

3. Graph

Mengapa belajar ini

Terma dan topik

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