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Penyelesaian - Absolute value equations

Hasil Akhir Langkah-langkah Terma dan topik

Exact form:

x = \frac{7}{2}, - \frac{21}{34}

x=\frac{7}{2} , -\frac{21}{34}

Mixed number form:

x = 3 \frac{1}{2}, - \frac{21}{34}

x=3\frac{1}{2} , -\frac{21}{34}

Decimal form:

x = 3.5, - 0.618

x=3.5 , -0.618

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
$| 4 x - \frac{4}{1} | = | \frac{6}{7} x + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - \frac{4}{1} \| = \| \frac{6}{7} x + 7 \|$
$x = + y$	$(4 x - \frac{4}{1}) = (\frac{6}{7} x + 7)$
$x = - y$	$(4 x - \frac{4}{1}) = - (\frac{6}{7} x + 7)$
$+ x = y$	$(4 x - \frac{4}{1}) = (\frac{6}{7} x + 7)$
$- x = y$	$- (4 x - \frac{4}{1}) = (\frac{6}{7} x + 7)$

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:

$\| x \| = \| y \|$	$\| 4 x - \frac{4}{1} \| = \| \frac{6}{7} x + 7 \|$
$x = + y$ , $+ x = y$	$(4 x - \frac{4}{1}) = (\frac{6}{7} x + 7)$
$x = - y$ , $- x = y$	$(4 x - \frac{4}{1}) = - (\frac{6}{7} x + 7)$

2. Solve the two equations for x

20 additional steps

$4 x + \frac{- 4}{1} = (\frac{6}{7} x + 7)$

A variable's value does not change when it is divided by 1, so we can eliminate it:

$4 x - 4 = (\frac{6}{7} x + 7)$

Subtract from both sides:

$(4 x - 4) - \frac{6}{7} \cdot x = (\frac{6}{7} x + 7) - \frac{6}{7} x$

Kumpulkan sebutan sejenis:

$(4 x + \frac{- 6}{7} \cdot x) - 4 = (\frac{6}{7} \cdot x + 7) - \frac{6}{7} x$

Kumpulkan pekali:

$(4 + \frac{- 6}{7}) x - 4 = (\frac{6}{7} \cdot x + 7) - \frac{6}{7} x$

Tukar nombor bulat kepada pecahan:

$(\frac{28}{7} + \frac{- 6}{7}) x - 4 = (\frac{6}{7} \cdot x + 7) - \frac{6}{7} x$

Gabungkan pecahan:

$\frac{(28 - 6)}{7} \cdot x - 4 = (\frac{6}{7} \cdot x + 7) - \frac{6}{7} x$

Gabungkan pembilang:

$\frac{22}{7} \cdot x - 4 = (\frac{6}{7} \cdot x + 7) - \frac{6}{7} x$

Kumpulkan sebutan sejenis:

$\frac{22}{7} \cdot x - 4 = (\frac{6}{7} \cdot x + \frac{- 6}{7} x) + 7$

Gabungkan pecahan:

$\frac{22}{7} \cdot x - 4 = \frac{(6 - 6)}{7} x + 7$

Gabungkan pembilang:

$\frac{22}{7} \cdot x - 4 = \frac{0}{7} x + 7$

Permudahkan pembilang sifar:

$\frac{22}{7} x - 4 = 0 x + 7$

Permudahkan aritmetik:

$\frac{22}{7} x - 4 = 7$

Add to both sides:

$(\frac{22}{7} x - 4) + 4 = 7 + 4$

Permudahkan aritmetik:

$\frac{22}{7} x = 7 + 4$

Permudahkan aritmetik:

$\frac{22}{7} x = 11$

Multiply both sides by inverse fraction :

$(\frac{22}{7} x) \cdot \frac{7}{22} = 11 \cdot \frac{7}{22}$

Kumpulkan sebutan sejenis:

$(\frac{22}{7} \cdot \frac{7}{22}) x = 11 \cdot \frac{7}{22}$

Darabkan pekali:

$\frac{(22 \cdot 7)}{(7 \cdot 22)} x = 11 \cdot \frac{7}{22}$

Permudahkan pecahan:

$x = 11 \cdot \frac{7}{22}$

Darabkan pecahan:

$x = \frac{(11 \cdot 7)}{22}$

Permudahkan aritmetik:

$x = \frac{7}{2}$

21 additional steps

$4 x + \frac{- 4}{1} = - (\frac{6}{7} x + 7)$

A variable's value does not change when it is divided by 1, so we can eliminate it:

$4 x - 4 = - (\frac{6}{7} x + 7)$

Kembangkan tanda kurung:

$4 x - 4 = \frac{- 6}{7} x - 7$

Add to both sides:

$(4 x - 4) + \frac{6}{7} \cdot x = (\frac{- 6}{7} x - 7) + \frac{6}{7} x$

Kumpulkan sebutan sejenis:

$(4 x + \frac{6}{7} \cdot x) - 4 = (\frac{- 6}{7} \cdot x - 7) + \frac{6}{7} x$

Kumpulkan pekali:

$(4 + \frac{6}{7}) x - 4 = (\frac{- 6}{7} \cdot x - 7) + \frac{6}{7} x$

Tukar nombor bulat kepada pecahan:

$(\frac{28}{7} + \frac{6}{7}) x - 4 = (\frac{- 6}{7} \cdot x - 7) + \frac{6}{7} x$

Gabungkan pecahan:

$\frac{(28 + 6)}{7} \cdot x - 4 = (\frac{- 6}{7} \cdot x - 7) + \frac{6}{7} x$

Gabungkan pembilang:

$\frac{34}{7} \cdot x - 4 = (\frac{- 6}{7} \cdot x - 7) + \frac{6}{7} x$

Kumpulkan sebutan sejenis:

$\frac{34}{7} \cdot x - 4 = (\frac{- 6}{7} \cdot x + \frac{6}{7} x) - 7$

Gabungkan pecahan:

$\frac{34}{7} \cdot x - 4 = \frac{(- 6 + 6)}{7} x - 7$

Gabungkan pembilang:

$\frac{34}{7} \cdot x - 4 = \frac{0}{7} x - 7$

Permudahkan pembilang sifar:

$\frac{34}{7} x - 4 = 0 x - 7$

Permudahkan aritmetik:

$\frac{34}{7} x - 4 = - 7$

Add to both sides:

$(\frac{34}{7} x - 4) + 4 = - 7 + 4$

Permudahkan aritmetik:

$\frac{34}{7} x = - 7 + 4$

Permudahkan aritmetik:

$\frac{34}{7} x = - 3$

Multiply both sides by inverse fraction :

$(\frac{34}{7} x) \cdot \frac{7}{34} = - 3 \cdot \frac{7}{34}$

Kumpulkan sebutan sejenis:

$(\frac{34}{7} \cdot \frac{7}{34}) x = - 3 \cdot \frac{7}{34}$

Darabkan pekali:

$\frac{(34 \cdot 7)}{(7 \cdot 34)} x = - 3 \cdot \frac{7}{34}$

Permudahkan pecahan:

$x = - 3 \cdot \frac{7}{34}$

Darabkan pecahan:

$x = \frac{(- 3 \cdot 7)}{34}$

Permudahkan aritmetik:

$x = \frac{- 21}{34}$

3. List the solutions

$x = \frac{7}{2}, - \frac{21}{34}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - \frac{4}{1} |$
$y = | \frac{6}{7} x + 7 |$
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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