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Solution - Absolute value equations

Exact form:

x = \frac{1}{5}, \frac{1}{9}

x=\frac{1}{5} , \frac{1}{9}

Decimal form:

x = 0.2, 0.111

x=0.2 , 0.111

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| 2 x | - | 7 x - 1 | = 0$

Add $| 7 x - 1 |$ to both sides of the equation:

$| 2 x | - | 7 x - 1 | + | 7 x - 1 | = | 7 x - 1 |$

Simplify the arithmetic

$| 2 x | = | 7 x - 1 |$

2. 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
$| 2 x | = | 7 x - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x \| = \| 7 x - 1 \|$
$x = + y$	$(2 x) = (7 x - 1)$
$x = - y$	$(2 x) = (- (7 x - 1))$
$+ x = y$	$(2 x) = (7 x - 1)$
$- x = y$	$- (2 x) = (7 x - 1)$

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:

$\| x \| = \| y \|$	$\| 2 x \| = \| 7 x - 1 \|$
$x = + y$ , $+ x = y$	$(2 x) = (7 x - 1)$
$x = - y$ , $- x = y$	$(2 x) = (- (7 x - 1))$

3. Solve the two equations for x

7 additional steps

$2 x = (7 x - 1)$

Subtract from both sides:

$(2 x) - 7 x = (7 x - 1) - 7 x$

Simplify the arithmetic:

$- 5 x = (7 x - 1) - 7 x$

Group like terms:

$- 5 x = (7 x - 7 x) - 1$

Simplify the arithmetic:

$- 5 x = - 1$

Divide both sides by :

$\frac{(- 5 x)}{- 5} = \frac{- 1}{- 5}$

Cancel out the negatives:

$\frac{5 x}{5} = \frac{- 1}{- 5}$

Simplify the fraction:

$x = \frac{- 1}{- 5}$

Cancel out the negatives:

$x = \frac{1}{5}$

6 additional steps

$2 x = - (7 x - 1)$

Expand the parentheses:

$2 x = - 7 x + 1$

Add to both sides:

$(2 x) + 7 x = (- 7 x + 1) + 7 x$

Simplify the arithmetic:

$9 x = (- 7 x + 1) + 7 x$

Group like terms:

$9 x = (- 7 x + 7 x) + 1$

Simplify the arithmetic:

$9 x = 1$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{1}{9}$

Simplify the fraction:

$x = \frac{1}{9}$

4. List the solutions

$x = \frac{1}{5}, \frac{1}{9}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 2 x |$
$y = | 7 x - 1 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

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