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Solution - Absolute value equations

Exact form:

x = - \frac{6}{5}, \frac{2}{3}

x=-\frac{6}{5} , \frac{2}{3}

Mixed number form:

x = - 1 \frac{1}{5}, \frac{2}{3}

x=-1\frac{1}{5} , \frac{2}{3}

Decimal form:

x = - 1.2, 0.667

x=-1.2 , 0.667

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
$| 2 x - 6 | = | 7 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 6 \| = \| 7 x \|$
$x = + y$	$(2 x - 6) = (7 x)$
$x = - y$	$(2 x - 6) = - (7 x)$
$+ x = y$	$(2 x - 6) = (7 x)$
$- x = y$	$- (2 x - 6) = (7 x)$

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:

$\| x \| = \| y \|$	$\| 2 x - 6 \| = \| 7 x \|$
$x = + y$ , $+ x = y$	$(2 x - 6) = (7 x)$
$x = - y$ , $- x = y$	$(2 x - 6) = - (7 x)$

2. Solve the two equations for x

10 additional steps

$(2 x - 6) = 7 x$

Subtract from both sides:

$(2 x - 6) - 7 x = (7 x) - 7 x$

Group like terms:

$(2 x - 7 x) - 6 = (7 x) - 7 x$

Simplify the arithmetic:

$- 5 x - 6 = (7 x) - 7 x$

Simplify the arithmetic:

$- 5 x - 6 = 0$

Add to both sides:

$(- 5 x - 6) + 6 = 0 + 6$

Simplify the arithmetic:

$- 5 x = 0 + 6$

Simplify the arithmetic:

$- 5 x = 6$

Divide both sides by :

$\frac{(- 5 x)}{- 5} = \frac{6}{- 5}$

Cancel out the negatives:

$\frac{5 x}{5} = \frac{6}{- 5}$

Simplify the fraction:

$x = \frac{6}{- 5}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 6}{5}$

9 additional steps

$(2 x - 6) = - 7 x$

Add to both sides:

$(2 x - 6) + 6 = (- 7 x) + 6$

Simplify the arithmetic:

$2 x = (- 7 x) + 6$

Add to both sides:

$(2 x) + 7 x = ((- 7 x) + 6) + 7 x$

Simplify the arithmetic:

$9 x = ((- 7 x) + 6) + 7 x$

Group like terms:

$9 x = (- 7 x + 7 x) + 6$

Simplify the arithmetic:

$9 x = 6$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{6}{9}$

Simplify the fraction:

$x = \frac{6}{9}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 3)}{(3 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{2}{3}$

3. List the solutions

$x = - \frac{6}{5}, \frac{2}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 6 |$
$y = | 7 x |$
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 x

3. List the solutions

4. 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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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