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Solution - Absolute value equations

Exact form:

i = \frac{9}{8}, - \frac{7}{6}

i=\frac{9}{8} , -\frac{7}{6}

Mixed number form:

i = 1 \frac{1}{8}, - 1 \frac{1}{6}

i=1\frac{1}{8} , -1\frac{1}{6}

Decimal form:

i = 1.125, - 1.167

i=1.125 , -1.167

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
$| - i + 8 | = | 7 i - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - i + 8 \| = \| 7 i - 1 \|$
$x = + y$	$(- i + 8) = (7 i - 1)$
$x = - y$	$(- i + 8) = - (7 i - 1)$
$+ x = y$	$(- i + 8) = (7 i - 1)$
$- x = y$	$- (- i + 8) = (7 i - 1)$

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:

$\| x \| = \| y \|$	$\| - i + 8 \| = \| 7 i - 1 \|$
$x = + y$ , $+ x = y$	$(- i + 8) = (7 i - 1)$
$x = - y$ , $- x = y$	$(- i + 8) = - (7 i - 1)$

2. Solve the two equations for i

11 additional steps

$(- i + 8) = (7 i - 1)$

Subtract from both sides:

$(- i + 8) - 7 i = (7 i - 1) - 7 i$

Group like terms:

$(- i - 7 i) + 8 = (7 i - 1) - 7 i$

Simplify the arithmetic:

$- 8 i + 8 = (7 i - 1) - 7 i$

Group like terms:

$- 8 i + 8 = (7 i - 7 i) - 1$

Simplify the arithmetic:

$- 8 i + 8 = - 1$

Subtract from both sides:

$(- 8 i + 8) - 8 = - 1 - 8$

Simplify the arithmetic:

$- 8 i = - 1 - 8$

Simplify the arithmetic:

$- 8 i = - 9$

Divide both sides by :

$\frac{(- 8 i)}{- 8} = \frac{- 9}{- 8}$

Cancel out the negatives:

$\frac{8 i}{8} = \frac{- 9}{- 8}$

Simplify the fraction:

$i = \frac{- 9}{- 8}$

Cancel out the negatives:

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

10 additional steps

$(- i + 8) = - (7 i - 1)$

Expand the parentheses:

$(- i + 8) = - 7 i + 1$

Add to both sides:

$(- i + 8) + 7 i = (- 7 i + 1) + 7 i$

Group like terms:

$(- i + 7 i) + 8 = (- 7 i + 1) + 7 i$

Simplify the arithmetic:

$6 i + 8 = (- 7 i + 1) + 7 i$

Group like terms:

$6 i + 8 = (- 7 i + 7 i) + 1$

Simplify the arithmetic:

$6 i + 8 = 1$

Subtract from both sides:

$(6 i + 8) - 8 = 1 - 8$

Simplify the arithmetic:

$6 i = 1 - 8$

Simplify the arithmetic:

$6 i = - 7$

Divide both sides by :

$\frac{(6 i)}{6} = \frac{- 7}{6}$

Simplify the fraction:

$i = \frac{- 7}{6}$

3. List the solutions

$i = \frac{9}{8}, - \frac{7}{6}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - i + 8 |$
$y = | 7 i - 1 |$
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 i

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 i

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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