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Solution - Absolute value equations

Exact form:

i = \frac{9}{13}, 3

i=\frac{9}{13} , 3

Decimal form:

i = 0.692, 3

i=0.692 , 3

See steps

Step-by-step explanation

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

$| 9 i - 12 | + | 4 i + 3 | = 0$

Add $- | 4 i + 3 |$ to both sides of the equation:

$| 9 i - 12 | + | 4 i + 3 | - | 4 i + 3 | = - | 4 i + 3 |$

Simplify the arithmetic

$| 9 i - 12 | = - | 4 i + 3 |$

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
$| 9 i - 12 | = - | 4 i + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 i - 12 \| = - \| 4 i + 3 \|$
$x = + y$	$(9 i - 12) = - (4 i + 3)$
$x = - y$	$(9 i - 12) = - - (4 i + 3)$
$+ x = y$	$(9 i - 12) = - (4 i + 3)$
$- x = y$	$- (9 i - 12) = - (4 i + 3)$

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:

$\| x \| = \| y \|$	$\| 9 i - 12 \| = - \| 4 i + 3 \|$
$x = + y$ , $+ x = y$	$(9 i - 12) = - (4 i + 3)$
$x = - y$ , $- x = y$	$(9 i - 12) = - - (4 i + 3)$

3. Solve the two equations for i

10 additional steps

$(9 i - 12) = - (4 i + 3)$

Expand the parentheses:

$(9 i - 12) = - 4 i - 3$

Add to both sides:

$(9 i - 12) + 4 i = (- 4 i - 3) + 4 i$

Group like terms:

$(9 i + 4 i) - 12 = (- 4 i - 3) + 4 i$

Simplify the arithmetic:

$13 i - 12 = (- 4 i - 3) + 4 i$

Group like terms:

$13 i - 12 = (- 4 i + 4 i) - 3$

Simplify the arithmetic:

$13 i - 12 = - 3$

Add to both sides:

$(13 i - 12) + 12 = - 3 + 12$

Simplify the arithmetic:

$13 i = - 3 + 12$

Simplify the arithmetic:

$13 i = 9$

Divide both sides by :

$\frac{(13 i)}{13} = \frac{9}{13}$

Simplify the fraction:

$i = \frac{9}{13}$

12 additional steps

$(9 i - 12) = - (- (4 i + 3))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(9 i - 12) = 4 i + 3$

Subtract from both sides:

$(9 i - 12) - 4 i = (4 i + 3) - 4 i$

Group like terms:

$(9 i - 4 i) - 12 = (4 i + 3) - 4 i$

Simplify the arithmetic:

$5 i - 12 = (4 i + 3) - 4 i$

Group like terms:

$5 i - 12 = (4 i - 4 i) + 3$

Simplify the arithmetic:

$5 i - 12 = 3$

Add to both sides:

$(5 i - 12) + 12 = 3 + 12$

Simplify the arithmetic:

$5 i = 3 + 12$

Simplify the arithmetic:

$5 i = 15$

Divide both sides by :

$\frac{(5 i)}{5} = \frac{15}{5}$

Simplify the fraction:

$i = \frac{15}{5}$

Find the greatest common factor of the numerator and denominator:

$i = \frac{(3 \cdot 5)}{(1 \cdot 5)}$

Factor out and cancel the greatest common factor:

$i = 3$

4. List the solutions

$i = \frac{9}{13}, 3$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 9 i - 12 |$
$y = - | 4 i + 3 |$
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