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Solution - Absolute value equations

Exact form:

i = - \frac{1}{22}, \frac{1}{28}

i=-\frac{1}{22} , \frac{1}{28}

Decimal form:

i = - 0.045, 0.036

i=-0.045 , 0.036

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| - 3 i + 1 | + | 25 i | = 0$

Add $- | 25 i |$ to both sides of the equation:

$| - 3 i + 1 | + | 25 i | - | 25 i | = - | 25 i |$

Simplify the arithmetic

$| - 3 i + 1 | = - | 25 i |$

2. 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
$| - 3 i + 1 | = - | 25 i |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 i + 1 \| = - \| 25 i \|$
$x = + y$	$(- 3 i + 1) = - (25 i)$
$x = - y$	$(- 3 i + 1) = - - (25 i)$
$+ x = y$	$(- 3 i + 1) = - (25 i)$
$- x = y$	$- (- 3 i + 1) = - (25 i)$

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:

$\| x \| = \| y \|$	$\| - 3 i + 1 \| = - \| 25 i \|$
$x = + y$ , $+ x = y$	$(- 3 i + 1) = - (25 i)$
$x = - y$ , $- x = y$	$(- 3 i + 1) = - - (25 i)$

3. Solve the two equations for i

7 additional steps

$(- 3 i + 1) = - 25 i$

Subtract from both sides:

$(- 3 i + 1) - 1 = (- 25 i) - 1$

Simplify the arithmetic:

$- 3 i = (- 25 i) - 1$

Add to both sides:

$(- 3 i) + 25 i = ((- 25 i) - 1) + 25 i$

Simplify the arithmetic:

$22 i = ((- 25 i) - 1) + 25 i$

Group like terms:

$22 i = (- 25 i + 25 i) - 1$

Simplify the arithmetic:

$22 i = - 1$

Divide both sides by :

$\frac{(22 i)}{22} = \frac{- 1}{22}$

Simplify the fraction:

$i = \frac{- 1}{22}$

12 additional steps

$(- 3 i + 1) = - - 25 i$

Group like terms:

$(- 3 i + 1) = (- 1 \cdot - 25) i$

Multiply the coefficients:

$(- 3 i + 1) = 25 i$

Subtract from both sides:

$(- 3 i + 1) - 25 i = (25 i) - 25 i$

Group like terms:

$(- 3 i - 25 i) + 1 = (25 i) - 25 i$

Simplify the arithmetic:

$- 28 i + 1 = (25 i) - 25 i$

Simplify the arithmetic:

$- 28 i + 1 = 0$

Subtract from both sides:

$(- 28 i + 1) - 1 = 0 - 1$

Simplify the arithmetic:

$- 28 i = 0 - 1$

Simplify the arithmetic:

$- 28 i = - 1$

Divide both sides by :

$\frac{(- 28 i)}{- 28} = \frac{- 1}{- 28}$

Cancel out the negatives:

$\frac{28 i}{28} = \frac{- 1}{- 28}$

Simplify the fraction:

$i = \frac{- 1}{- 28}$

Cancel out the negatives:

$i = \frac{1}{28}$

4. List the solutions

$i = - \frac{1}{22}, \frac{1}{28}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - 3 i + 1 |$
$y = - | 25 i |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for i

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for i

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved