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Solution - Absolute value equations

Exact form:

i = - \frac{3}{2}, - \frac{1}{8}

i=-\frac{3}{2} , -\frac{1}{8}

Mixed number form:

i = - 1 \frac{1}{2}, - \frac{1}{8}

i=-1\frac{1}{2} , -\frac{1}{8}

Decimal form:

i = - 1.5, - 0.125

i=-1.5 , -0.125

See steps

Step-by-step explanation

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

$| - 3 i + 1 | + | 5 i + 2 | = 0$

Add $- | 5 i + 2 |$ to both sides of the equation:

$| - 3 i + 1 | + | 5 i + 2 | - | 5 i + 2 | = - | 5 i + 2 |$

Simplify the arithmetic

$| - 3 i + 1 | = - | 5 i + 2 |$

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
$| - 3 i + 1 | = - | 5 i + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 i + 1 \| = - \| 5 i + 2 \|$
$x = + y$	$(- 3 i + 1) = - (5 i + 2)$
$x = - y$	$(- 3 i + 1) = - - (5 i + 2)$
$+ x = y$	$(- 3 i + 1) = - (5 i + 2)$
$- x = y$	$- (- 3 i + 1) = - (5 i + 2)$

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:

$\| x \| = \| y \|$	$\| - 3 i + 1 \| = - \| 5 i + 2 \|$
$x = + y$ , $+ x = y$	$(- 3 i + 1) = - (5 i + 2)$
$x = - y$ , $- x = y$	$(- 3 i + 1) = - - (5 i + 2)$

3. Solve the two equations for i

10 additional steps

$(- 3 i + 1) = - (5 i + 2)$

Expand the parentheses:

$(- 3 i + 1) = - 5 i - 2$

Add to both sides:

$(- 3 i + 1) + 5 i = (- 5 i - 2) + 5 i$

Group like terms:

$(- 3 i + 5 i) + 1 = (- 5 i - 2) + 5 i$

Simplify the arithmetic:

$2 i + 1 = (- 5 i - 2) + 5 i$

Group like terms:

$2 i + 1 = (- 5 i + 5 i) - 2$

Simplify the arithmetic:

$2 i + 1 = - 2$

Subtract from both sides:

$(2 i + 1) - 1 = - 2 - 1$

Simplify the arithmetic:

$2 i = - 2 - 1$

Simplify the arithmetic:

$2 i = - 3$

Divide both sides by :

$\frac{(2 i)}{2} = \frac{- 3}{2}$

Simplify the fraction:

$i = \frac{- 3}{2}$

12 additional steps

$(- 3 i + 1) = - (- (5 i + 2))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(- 3 i + 1) = 5 i + 2$

Subtract from both sides:

$(- 3 i + 1) - 5 i = (5 i + 2) - 5 i$

Group like terms:

$(- 3 i - 5 i) + 1 = (5 i + 2) - 5 i$

Simplify the arithmetic:

$- 8 i + 1 = (5 i + 2) - 5 i$

Group like terms:

$- 8 i + 1 = (5 i - 5 i) + 2$

Simplify the arithmetic:

$- 8 i + 1 = 2$

Subtract from both sides:

$(- 8 i + 1) - 1 = 2 - 1$

Simplify the arithmetic:

$- 8 i = 2 - 1$

Simplify the arithmetic:

$- 8 i = 1$

Divide both sides by :

$\frac{(- 8 i)}{- 8} = \frac{1}{- 8}$

Cancel out the negatives:

$\frac{8 i}{8} = \frac{1}{- 8}$

Simplify the fraction:

$i = \frac{1}{- 8}$

Move the negative sign from the denominator to the numerator:

$i = \frac{- 1}{8}$

4. List the solutions

$i = - \frac{3}{2}, - \frac{1}{8}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - 3 i + 1 |$
$y = - | 5 i + 2 |$
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