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Penyelesaian - Absolute value equations

Hasil Akhir Langkah-langkah Terma dan topik

Exact form:

x = 1, \frac{1}{7}

x=1 , \frac{1}{7}

Decimal form:

x = 1, 0.143

x=1 , 0.143

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
$| - 8 x + 5 | = | x - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 8 x + 5 \| = \| x - 4 \|$
$x = + y$	$(- 8 x + 5) = (x - 4)$
$x = - y$	$(- 8 x + 5) = - (x - 4)$
$+ x = y$	$(- 8 x + 5) = (x - 4)$
$- x = y$	$- (- 8 x + 5) = (x - 4)$

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:

$\| x \| = \| y \|$	$\| - 8 x + 5 \| = \| x - 4 \|$
$x = + y$ , $+ x = y$	$(- 8 x + 5) = (x - 4)$
$x = - y$ , $- x = y$	$(- 8 x + 5) = - (x - 4)$

2. Solve the two equations for x

12 additional steps

$(- 8 x + 5) = (x - 4)$

Subtract from both sides:

$(- 8 x + 5) - x = (x - 4) - x$

Kumpulkan sebutan sejenis:

$(- 8 x - x) + 5 = (x - 4) - x$

Permudahkan aritmetik:

$- 9 x + 5 = (x - 4) - x$

Kumpulkan sebutan sejenis:

$- 9 x + 5 = (x - x) - 4$

Permudahkan aritmetik:

$- 9 x + 5 = - 4$

Subtract from both sides:

$(- 9 x + 5) - 5 = - 4 - 5$

Permudahkan aritmetik:

$- 9 x = - 4 - 5$

Permudahkan aritmetik:

$- 9 x = - 9$

Divide both sides by :

$\frac{(- 9 x)}{- 9} = \frac{- 9}{- 9}$

Hapuskan tanda negatif:

$\frac{9 x}{9} = \frac{- 9}{- 9}$

Permudahkan pecahan:

$x = \frac{- 9}{- 9}$

Hapuskan tanda negatif:

$x = \frac{9}{9}$

Permudahkan pecahan:

$x = 1$

12 additional steps

$(- 8 x + 5) = - (x - 4)$

Expand the parentheses:

$(- 8 x + 5) = - x + 4$

Add to both sides:

$(- 8 x + 5) + x = (- x + 4) + x$

Kumpulkan sebutan sejenis:

$(- 8 x + x) + 5 = (- x + 4) + x$

Permudahkan aritmetik:

$- 7 x + 5 = (- x + 4) + x$

Kumpulkan sebutan sejenis:

$- 7 x + 5 = (- x + x) + 4$

Permudahkan aritmetik:

$- 7 x + 5 = 4$

Subtract from both sides:

$(- 7 x + 5) - 5 = 4 - 5$

Permudahkan aritmetik:

$- 7 x = 4 - 5$

Permudahkan aritmetik:

$- 7 x = - 1$

Divide both sides by :

$\frac{(- 7 x)}{- 7} = \frac{- 1}{- 7}$

Hapuskan tanda negatif:

$\frac{7 x}{7} = \frac{- 1}{- 7}$

Permudahkan pecahan:

$x = \frac{- 1}{- 7}$

Hapuskan tanda negatif:

$x = \frac{1}{7}$

3. List the solutions

$x = 1, \frac{1}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 8 x + 5 |$
$y = | x - 4 |$
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 x

3. List the solutions

4. Graph

Mengapa belajar ini

Terma dan topik

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