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Solution - Absolute value equations

Exact form:

x = 2, - \frac{8}{3}

x=2 , -\frac{8}{3}

Mixed number form:

x = 2, - 2 \frac{2}{3}

x=2 , -2\frac{2}{3}

Decimal form:

x = 2, - 2.667

x=2 , -2.667

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
$| 4 x - 1 | = | - x + 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 1 \| = \| - x + 9 \|$
$x = + y$	$(4 x - 1) = (- x + 9)$
$x = - y$	$(4 x - 1) = - (- x + 9)$
$+ x = y$	$(4 x - 1) = (- x + 9)$
$- x = y$	$- (4 x - 1) = (- x + 9)$

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:

$\| x \| = \| y \|$	$\| 4 x - 1 \| = \| - x + 9 \|$
$x = + y$ , $+ x = y$	$(4 x - 1) = (- x + 9)$
$x = - y$ , $- x = y$	$(4 x - 1) = - (- x + 9)$

2. Solve the two equations for x

11 additional steps

$(4 x - 1) = (- x + 9)$

Add to both sides:

$(4 x - 1) + x = (- x + 9) + x$

Group like terms:

$(4 x + x) - 1 = (- x + 9) + x$

Simplify the arithmetic:

$5 x - 1 = (- x + 9) + x$

Group like terms:

$5 x - 1 = (- x + x) + 9$

Simplify the arithmetic:

$5 x - 1 = 9$

Add to both sides:

$(5 x - 1) + 1 = 9 + 1$

Simplify the arithmetic:

$5 x = 9 + 1$

Simplify the arithmetic:

$5 x = 10$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{10}{5}$

Simplify the fraction:

$x = \frac{10}{5}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 5)}{(1 \cdot 5)}$

Factor out and cancel the greatest common factor:

$x = 2$

10 additional steps

$(4 x - 1) = - (- x + 9)$

Expand the parentheses:

$(4 x - 1) = x - 9$

Subtract from both sides:

$(4 x - 1) - x = (x - 9) - x$

Group like terms:

$(4 x - x) - 1 = (x - 9) - x$

Simplify the arithmetic:

$3 x - 1 = (x - 9) - x$

Group like terms:

$3 x - 1 = (x - x) - 9$

Simplify the arithmetic:

$3 x - 1 = - 9$

Add to both sides:

$(3 x - 1) + 1 = - 9 + 1$

Simplify the arithmetic:

$3 x = - 9 + 1$

Simplify the arithmetic:

$3 x = - 8$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{- 8}{3}$

Simplify the fraction:

$x = \frac{- 8}{3}$

3. List the solutions

$x = 2, - \frac{8}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 1 |$
$y = | - x + 9 |$
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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