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Solution - Absolute value equations

Exact form:

i = \frac{12}{43}, - \frac{12}{25}

i=\frac{12}{43} , -\frac{12}{25}

Decimal form:

i = 0.279, - 0.48

i=0.279 , -0.48

See steps

Step-by-step explanation

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

$| 9 i - 12 | + | 34 i | = 0$

Add $- | 34 i |$ to both sides of the equation:

$| 9 i - 12 | + | 34 i | - | 34 i | = - | 34 i |$

Simplify the arithmetic

$| 9 i - 12 | = - | 34 i |$

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
$| 9 i - 12 | = - | 34 i |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 i - 12 \| = - \| 34 i \|$
$x = + y$	$(9 i - 12) = - (34 i)$
$x = - y$	$(9 i - 12) = - - (34 i)$
$+ x = y$	$(9 i - 12) = - (34 i)$
$- x = y$	$- (9 i - 12) = - (34 i)$

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:

$\| x \| = \| y \|$	$\| 9 i - 12 \| = - \| 34 i \|$
$x = + y$ , $+ x = y$	$(9 i - 12) = - (34 i)$
$x = - y$ , $- x = y$	$(9 i - 12) = - - (34 i)$

3. Solve the two equations for i

7 additional steps

$(9 i - 12) = - 34 i$

Add to both sides:

$(9 i - 12) + 12 = (- 34 i) + 12$

Simplify the arithmetic:

$9 i = (- 34 i) + 12$

Add to both sides:

$(9 i) + 34 i = ((- 34 i) + 12) + 34 i$

Simplify the arithmetic:

$43 i = ((- 34 i) + 12) + 34 i$

Group like terms:

$43 i = (- 34 i + 34 i) + 12$

Simplify the arithmetic:

$43 i = 12$

Divide both sides by :

$\frac{(43 i)}{43} = \frac{12}{43}$

Simplify the fraction:

$i = \frac{12}{43}$

12 additional steps

$(9 i - 12) = - - 34 i$

Group like terms:

$(9 i - 12) = (- 1 \cdot - 34) i$

Multiply the coefficients:

$(9 i - 12) = 34 i$

Subtract from both sides:

$(9 i - 12) - 34 i = (34 i) - 34 i$

Group like terms:

$(9 i - 34 i) - 12 = (34 i) - 34 i$

Simplify the arithmetic:

$- 25 i - 12 = (34 i) - 34 i$

Simplify the arithmetic:

$- 25 i - 12 = 0$

Add to both sides:

$(- 25 i - 12) + 12 = 0 + 12$

Simplify the arithmetic:

$- 25 i = 0 + 12$

Simplify the arithmetic:

$- 25 i = 12$

Divide both sides by :

$\frac{(- 25 i)}{- 25} = \frac{12}{- 25}$

Cancel out the negatives:

$\frac{25 i}{25} = \frac{12}{- 25}$

Simplify the fraction:

$i = \frac{12}{- 25}$

Move the negative sign from the denominator to the numerator:

$i = \frac{- 12}{25}$

4. List the solutions

$i = \frac{12}{43}, - \frac{12}{25}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 9 i - 12 |$
$y = - | 34 i |$
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