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Solution - Absolute value equations

Exact form:

x = \frac{5}{4}, \frac{1}{4}

x=\frac{5}{4} , \frac{1}{4}

Mixed number form:

x = 1 \frac{1}{4}, \frac{1}{4}

x=1\frac{1}{4} , \frac{1}{4}

Decimal form:

x = 1.25, 0.25

x=1.25 , 0.25

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
$| 4 x + 1 | = | 8 x - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x + 1 \| = \| 8 x - 4 \|$
$x = + y$	$(4 x + 1) = (8 x - 4)$
$x = - y$	$(4 x + 1) = - (8 x - 4)$
$+ x = y$	$(4 x + 1) = (8 x - 4)$
$- x = y$	$- (4 x + 1) = (8 x - 4)$

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:

$\| x \| = \| y \|$	$\| 4 x + 1 \| = \| 8 x - 4 \|$
$x = + y$ , $+ x = y$	$(4 x + 1) = (8 x - 4)$
$x = - y$ , $- x = y$	$(4 x + 1) = - (8 x - 4)$

2. Solve the two equations for x

11 additional steps

$(4 x + 1) = (8 x - 4)$

Subtract from both sides:

$(4 x + 1) - 8 x = (8 x - 4) - 8 x$

Group like terms:

$(4 x - 8 x) + 1 = (8 x - 4) - 8 x$

Simplify the arithmetic:

$- 4 x + 1 = (8 x - 4) - 8 x$

Group like terms:

$- 4 x + 1 = (8 x - 8 x) - 4$

Simplify the arithmetic:

$- 4 x + 1 = - 4$

Subtract from both sides:

$(- 4 x + 1) - 1 = - 4 - 1$

Simplify the arithmetic:

$- 4 x = - 4 - 1$

Simplify the arithmetic:

$- 4 x = - 5$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 5}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 5}{- 4}$

Simplify the fraction:

$x = \frac{- 5}{- 4}$

Cancel out the negatives:

$x = \frac{5}{4}$

12 additional steps

$(4 x + 1) = - (8 x - 4)$

Expand the parentheses:

$(4 x + 1) = - 8 x + 4$

Add to both sides:

$(4 x + 1) + 8 x = (- 8 x + 4) + 8 x$

Group like terms:

$(4 x + 8 x) + 1 = (- 8 x + 4) + 8 x$

Simplify the arithmetic:

$12 x + 1 = (- 8 x + 4) + 8 x$

Group like terms:

$12 x + 1 = (- 8 x + 8 x) + 4$

Simplify the arithmetic:

$12 x + 1 = 4$

Subtract from both sides:

$(12 x + 1) - 1 = 4 - 1$

Simplify the arithmetic:

$12 x = 4 - 1$

Simplify the arithmetic:

$12 x = 3$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{3}{12}$

Simplify the fraction:

$x = \frac{3}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 3)}{(4 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{1}{4}$

3. List the solutions

$x = \frac{5}{4}, \frac{1}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x + 1 |$
$y = | 8 x - 4 |$
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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