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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

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

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

Decimal form:

x = 2.5, - 0.4

x=2.5 , -0.4

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

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

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:

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

2. Solve the two equations for x

11 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$4 x - 3 = (3 x + 7) - 3 x$

Group like terms:

$4 x - 3 = (3 x - 3 x) + 7$

Simplify the arithmetic:

$4 x - 3 = 7$

Add to both sides:

$(4 x - 3) + 3 = 7 + 3$

Simplify the arithmetic:

$4 x = 7 + 3$

Simplify the arithmetic:

$4 x = 10$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{10}{4}$

Simplify the fraction:

$x = \frac{10}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

12 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$10 x - 3 = (- 3 x - 7) + 3 x$

Group like terms:

$10 x - 3 = (- 3 x + 3 x) - 7$

Simplify the arithmetic:

$10 x - 3 = - 7$

Add to both sides:

$(10 x - 3) + 3 = - 7 + 3$

Simplify the arithmetic:

$10 x = - 7 + 3$

Simplify the arithmetic:

$10 x = - 4$

Divide both sides by :

$\frac{(10 x)}{10} = \frac{- 4}{10}$

Simplify the fraction:

$x = \frac{- 4}{10}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

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

4. Graph

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