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Solution - Absolute value equations

Exact form:

x = \frac{7}{6}, \frac{5}{2}

x=\frac{7}{6} , \frac{5}{2}

Mixed number form:

x = 1 \frac{1}{6}, 2 \frac{1}{2}

x=1\frac{1}{6} , 2\frac{1}{2}

Decimal form:

x = 1.167, 2.5

x=1.167 , 2.5

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
$2 | 4 x - 8 | = | 2 x - 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| 4 x - 8 \| = \| 2 x - 9 \|$
$x = + y$	$2 (4 x - 8) = (2 x - 9)$
$x = - y$	$2 (4 x - 8) = - (2 x - 9)$
$+ x = y$	$2 (4 x - 8) = (2 x - 9)$
$- x = y$	$2 (- (4 x - 8)) = (2 x - 9)$

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:

$\| x \| = \| y \|$	$2 \| 4 x - 8 \| = \| 2 x - 9 \|$
$x = + y$ , $+ x = y$	$2 (4 x - 8) = (2 x - 9)$
$x = - y$ , $- x = y$	$2 (4 x - 8) = - (2 x - 9)$

2. Solve the two equations for x

12 additional steps

$2 \cdot (4 x - 8) = (2 x - 9)$

Expand the parentheses:

$2 \cdot 4 x + 2 \cdot - 8 = (2 x - 9)$

Multiply the coefficients:

$8 x + 2 \cdot - 8 = (2 x - 9)$

Simplify the arithmetic:

$8 x - 16 = (2 x - 9)$

Subtract from both sides:

$(8 x - 16) - 2 x = (2 x - 9) - 2 x$

Group like terms:

$(8 x - 2 x) - 16 = (2 x - 9) - 2 x$

Simplify the arithmetic:

$6 x - 16 = (2 x - 9) - 2 x$

Group like terms:

$6 x - 16 = (2 x - 2 x) - 9$

Simplify the arithmetic:

$6 x - 16 = - 9$

Add to both sides:

$(6 x - 16) + 16 = - 9 + 16$

Simplify the arithmetic:

$6 x = - 9 + 16$

Simplify the arithmetic:

$6 x = 7$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{7}{6}$

Simplify the fraction:

$x = \frac{7}{6}$

15 additional steps

$2 \cdot (4 x - 8) = - (2 x - 9)$

Expand the parentheses:

$2 \cdot 4 x + 2 \cdot - 8 = - (2 x - 9)$

Multiply the coefficients:

$8 x + 2 \cdot - 8 = - (2 x - 9)$

Simplify the arithmetic:

$8 x - 16 = - (2 x - 9)$

Expand the parentheses:

$8 x - 16 = - 2 x + 9$

Add to both sides:

$(8 x - 16) + 2 x = (- 2 x + 9) + 2 x$

Group like terms:

$(8 x + 2 x) - 16 = (- 2 x + 9) + 2 x$

Simplify the arithmetic:

$10 x - 16 = (- 2 x + 9) + 2 x$

Group like terms:

$10 x - 16 = (- 2 x + 2 x) + 9$

Simplify the arithmetic:

$10 x - 16 = 9$

Add to both sides:

$(10 x - 16) + 16 = 9 + 16$

Simplify the arithmetic:

$10 x = 9 + 16$

Simplify the arithmetic:

$10 x = 25$

Divide both sides by :

$\frac{(10 x)}{10} = \frac{25}{10}$

Simplify the fraction:

$x = \frac{25}{10}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

$x = \frac{7}{6}, \frac{5}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 2 | 4 x - 8 |$
$y = | 2 x - 9 |$
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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