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Penyelesaian - Absolute value equations

Hasil Akhir Langkah-langkah Terma dan topik

Exact form:

x = \frac{7}{4}

x=\frac{7}{4}

Mixed number form:

x = 1 \frac{3}{4}

x=1\frac{3}{4}

Decimal form:

x = 1.75

x=1.75

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
$| 2 x - 1 | = 2 | - x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = 2 \| - x + 3 \|$
$x = + y$	$(2 x - 1) = 2 (- x + 3)$
$x = - y$	$(2 x - 1) = 2 (- (- x + 3))$
$+ x = y$	$(2 x - 1) = 2 (- x + 3)$
$- x = y$	$- (2 x - 1) = 2 (- x + 3)$

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:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = 2 \| - x + 3 \|$
$x = + y$ , $+ x = y$	$(2 x - 1) = 2 (- x + 3)$
$x = - y$ , $- x = y$	$(2 x - 1) = 2 (- (- x + 3))$

2. Solve the two equations for x

13 additional steps

$(2 x - 1) = 2 \cdot (- x + 3)$

Expand the parentheses:

$(2 x - 1) = 2 \cdot - x + 2 \cdot 3$

Kumpulkan sebutan sejenis:

$(2 x - 1) = (2 \cdot - 1) x + 2 \cdot 3$

Darabkan pekali:

$(2 x - 1) = - 2 x + 2 \cdot 3$

Permudahkan aritmetik:

$(2 x - 1) = - 2 x + 6$

Add to both sides:

$(2 x - 1) + 2 x = (- 2 x + 6) + 2 x$

Kumpulkan sebutan sejenis:

$(2 x + 2 x) - 1 = (- 2 x + 6) + 2 x$

Permudahkan aritmetik:

$4 x - 1 = (- 2 x + 6) + 2 x$

Kumpulkan sebutan sejenis:

$4 x - 1 = (- 2 x + 2 x) + 6$

Permudahkan aritmetik:

$4 x - 1 = 6$

Add to both sides:

$(4 x - 1) + 1 = 6 + 1$

Permudahkan aritmetik:

$4 x = 6 + 1$

Permudahkan aritmetik:

$4 x = 7$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{7}{4}$

Permudahkan pecahan:

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

8 additional steps

$(2 x - 1) = 2 \cdot (- (- x + 3))$

Expand the parentheses:

$(2 x - 1) = 2 \cdot (x - 3)$

$(2 x - 1) = 2 x + 2 \cdot - 3$

Permudahkan aritmetik:

$(2 x - 1) = 2 x - 6$

Subtract from both sides:

$(2 x - 1) - 2 x = (2 x - 6) - 2 x$

Kumpulkan sebutan sejenis:

$(2 x - 2 x) - 1 = (2 x - 6) - 2 x$

Permudahkan aritmetik:

$- 1 = (2 x - 6) - 2 x$

Kumpulkan sebutan sejenis:

$- 1 = (2 x - 2 x) - 6$

Permudahkan aritmetik:

$- 1 = - 6$

Pernyataan ini palsu:

$- 1 = - 6$

The equation is false so it has no solution.

3. List the solutions

$x = \frac{7}{4}$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 1 |$
$y = 2 | - x + 3 |$
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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