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Solution - Absolute value equations

Exact form:

x = - \frac{1}{3}, \frac{6}{5}

x=-\frac{1}{3} , \frac{6}{5}

Mixed number form:

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

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

Decimal form:

x = - 0.333, 1.2

x=-0.333 , 1.2

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
$| 2 x - 7 | = | 8 x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 7 \| = \| 8 x - 5 \|$
$x = + y$	$(2 x - 7) = (8 x - 5)$
$x = - y$	$(2 x - 7) = - (8 x - 5)$
$+ x = y$	$(2 x - 7) = (8 x - 5)$
$- x = y$	$- (2 x - 7) = (8 x - 5)$

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:

$\| x \| = \| y \|$	$\| 2 x - 7 \| = \| 8 x - 5 \|$
$x = + y$ , $+ x = y$	$(2 x - 7) = (8 x - 5)$
$x = - y$ , $- x = y$	$(2 x - 7) = - (8 x - 5)$

2. Solve the two equations for x

13 additional steps

$(2 x - 7) = (8 x - 5)$

Subtract from both sides:

$(2 x - 7) - 8 x = (8 x - 5) - 8 x$

Group like terms:

$(2 x - 8 x) - 7 = (8 x - 5) - 8 x$

Simplify the arithmetic:

$- 6 x - 7 = (8 x - 5) - 8 x$

Group like terms:

$- 6 x - 7 = (8 x - 8 x) - 5$

Simplify the arithmetic:

$- 6 x - 7 = - 5$

Add to both sides:

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

Simplify the arithmetic:

$- 6 x = - 5 + 7$

Simplify the arithmetic:

$- 6 x = 2$

Divide both sides by :

$\frac{(- 6 x)}{- 6} = \frac{2}{- 6}$

Cancel out the negatives:

$\frac{6 x}{6} = \frac{2}{- 6}$

Simplify the fraction:

$x = \frac{2}{- 6}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 2}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{3}$

12 additional steps

$(2 x - 7) = - (8 x - 5)$

Expand the parentheses:

$(2 x - 7) = - 8 x + 5$

Add to both sides:

$(2 x - 7) + 8 x = (- 8 x + 5) + 8 x$

Group like terms:

$(2 x + 8 x) - 7 = (- 8 x + 5) + 8 x$

Simplify the arithmetic:

$10 x - 7 = (- 8 x + 5) + 8 x$

Group like terms:

$10 x - 7 = (- 8 x + 8 x) + 5$

Simplify the arithmetic:

$10 x - 7 = 5$

Add to both sides:

$(10 x - 7) + 7 = 5 + 7$

Simplify the arithmetic:

$10 x = 5 + 7$

Simplify the arithmetic:

$10 x = 12$

Divide both sides by :

$\frac{(10 x)}{10} = \frac{12}{10}$

Simplify the fraction:

$x = \frac{12}{10}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(6 \cdot 2)}{(5 \cdot 2)}$

Factor out and cancel the greatest common factor:

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

3. List the solutions

$x = - \frac{1}{3}, \frac{6}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 7 |$
$y = | 8 x - 5 |$
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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