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Solution - Absolute value equations

Exact form:

i = - \frac{1}{13}, \frac{3}{47}

i=-\frac{1}{13} , \frac{3}{47}

Decimal form:

i = - 0.077, 0.064

i=-0.077 , 0.064

See steps

Step-by-step explanation

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

$| - 4 i + 3 | + | 43 i | = 0$

Add $- | 43 i |$ to both sides of the equation:

$| - 4 i + 3 | + | 43 i | - | 43 i | = - | 43 i |$

Simplify the arithmetic

$| - 4 i + 3 | = - | 43 i |$

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
$| - 4 i + 3 | = - | 43 i |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 4 i + 3 \| = - \| 43 i \|$
$x = + y$	$(- 4 i + 3) = - (43 i)$
$x = - y$	$(- 4 i + 3) = - - (43 i)$
$+ x = y$	$(- 4 i + 3) = - (43 i)$
$- x = y$	$- (- 4 i + 3) = - (43 i)$

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:

$\| x \| = \| y \|$	$\| - 4 i + 3 \| = - \| 43 i \|$
$x = + y$ , $+ x = y$	$(- 4 i + 3) = - (43 i)$
$x = - y$ , $- x = y$	$(- 4 i + 3) = - - (43 i)$

3. Solve the two equations for i

9 additional steps

$(- 4 i + 3) = - 43 i$

Subtract from both sides:

$(- 4 i + 3) - 3 = (- 43 i) - 3$

Simplify the arithmetic:

$- 4 i = (- 43 i) - 3$

Add to both sides:

$(- 4 i) + 43 i = ((- 43 i) - 3) + 43 i$

Simplify the arithmetic:

$39 i = ((- 43 i) - 3) + 43 i$

Group like terms:

$39 i = (- 43 i + 43 i) - 3$

Simplify the arithmetic:

$39 i = - 3$

Divide both sides by :

$\frac{(39 i)}{39} = \frac{- 3}{39}$

Simplify the fraction:

$i = \frac{- 3}{39}$

Find the greatest common factor of the numerator and denominator:

$i = \frac{(- 1 \cdot 3)}{(13 \cdot 3)}$

Factor out and cancel the greatest common factor:

$i = \frac{- 1}{13}$

12 additional steps

$(- 4 i + 3) = - - 43 i$

Group like terms:

$(- 4 i + 3) = (- 1 \cdot - 43) i$

Multiply the coefficients:

$(- 4 i + 3) = 43 i$

Subtract from both sides:

$(- 4 i + 3) - 43 i = (43 i) - 43 i$

Group like terms:

$(- 4 i - 43 i) + 3 = (43 i) - 43 i$

Simplify the arithmetic:

$- 47 i + 3 = (43 i) - 43 i$

Simplify the arithmetic:

$- 47 i + 3 = 0$

Subtract from both sides:

$(- 47 i + 3) - 3 = 0 - 3$

Simplify the arithmetic:

$- 47 i = 0 - 3$

Simplify the arithmetic:

$- 47 i = - 3$

Divide both sides by :

$\frac{(- 47 i)}{- 47} = \frac{- 3}{- 47}$

Cancel out the negatives:

$\frac{47 i}{47} = \frac{- 3}{- 47}$

Simplify the fraction:

$i = \frac{- 3}{- 47}$

Cancel out the negatives:

$i = \frac{3}{47}$

4. List the solutions

$i = - \frac{1}{13}, \frac{3}{47}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - 4 i + 3 |$
$y = - | 43 i |$
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