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Solution - Absolute value equations

Exact form:

x = \frac{2}{5}, \frac{8}{9}

x=\frac{2}{5} , \frac{8}{9}

Decimal form:

x = 0.4, 0.889

x=0.4 , 0.889

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
$| 2 x - 3 | = | 7 x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 3 \| = \| 7 x - 5 \|$
$x = + y$	$(2 x - 3) = (7 x - 5)$
$x = - y$	$(2 x - 3) = - (7 x - 5)$
$+ x = y$	$(2 x - 3) = (7 x - 5)$
$- x = y$	$- (2 x - 3) = (7 x - 5)$

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:

$\| x \| = \| y \|$	$\| 2 x - 3 \| = \| 7 x - 5 \|$
$x = + y$ , $+ x = y$	$(2 x - 3) = (7 x - 5)$
$x = - y$ , $- x = y$	$(2 x - 3) = - (7 x - 5)$

2. Solve the two equations for x

11 additional steps

$(2 x - 3) = (7 x - 5)$

Subtract from both sides:

$(2 x - 3) - 7 x = (7 x - 5) - 7 x$

Group like terms:

$(2 x - 7 x) - 3 = (7 x - 5) - 7 x$

Simplify the arithmetic:

$- 5 x - 3 = (7 x - 5) - 7 x$

Group like terms:

$- 5 x - 3 = (7 x - 7 x) - 5$

Simplify the arithmetic:

$- 5 x - 3 = - 5$

Add to both sides:

$(- 5 x - 3) + 3 = - 5 + 3$

Simplify the arithmetic:

$- 5 x = - 5 + 3$

Simplify the arithmetic:

$- 5 x = - 2$

Divide both sides by :

$\frac{(- 5 x)}{- 5} = \frac{- 2}{- 5}$

Cancel out the negatives:

$\frac{5 x}{5} = \frac{- 2}{- 5}$

Simplify the fraction:

$x = \frac{- 2}{- 5}$

Cancel out the negatives:

$x = \frac{2}{5}$

10 additional steps

$(2 x - 3) = - (7 x - 5)$

Expand the parentheses:

$(2 x - 3) = - 7 x + 5$

Add to both sides:

$(2 x - 3) + 7 x = (- 7 x + 5) + 7 x$

Group like terms:

$(2 x + 7 x) - 3 = (- 7 x + 5) + 7 x$

Simplify the arithmetic:

$9 x - 3 = (- 7 x + 5) + 7 x$

Group like terms:

$9 x - 3 = (- 7 x + 7 x) + 5$

Simplify the arithmetic:

$9 x - 3 = 5$

Add to both sides:

$(9 x - 3) + 3 = 5 + 3$

Simplify the arithmetic:

$9 x = 5 + 3$

Simplify the arithmetic:

$9 x = 8$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{8}{9}$

Simplify the fraction:

$x = \frac{8}{9}$

3. List the solutions

$x = \frac{2}{5}, \frac{8}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 3 |$
$y = | 7 x - 5 |$
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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