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Solution - Absolute value equations

Exact form:

o = 0, 0

o=0 , 0

See steps

Step-by-step explanation

1. 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
$7 | \frac{1}{4} o | = | 4 o |$
without the absolute value bars:

$\| x \| = \| y \|$	$7 \| \frac{1}{4} o \| = \| 4 o \|$
$x = + y$	$7 (\frac{1}{4} o) = (4 o)$
$x = - y$	$7 (\frac{1}{4} o) = - (4 o)$
$+ x = y$	$7 (\frac{1}{4} o) = (4 o)$
$- x = y$	$7 (- (\frac{1}{4} o)) = (4 o)$

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:

$\| x \| = \| y \|$	$7 \| \frac{1}{4} o \| = \| 4 o \|$
$x = + y$ , $+ x = y$	$7 (\frac{1}{4} o) = (4 o)$
$x = - y$ , $- x = y$	$7 (\frac{1}{4} o) = - (4 o)$

2. Solve the two equations for o

8 additional steps

$7 \cdot \frac{1}{4} o = 4 o$

Multiply the coefficients:

$\frac{(7 \cdot 1)}{4} o = 4 o$

Combine like terms:

$\frac{7}{4} o = 4 o$

Subtract from both sides:

$(\frac{7}{4} o) - 4 o = (4 o) - 4 o$

Group the coefficients:

$(\frac{7}{4} - 4) o = (4 o) - 4 o$

Convert the integer into a fraction:

$(\frac{7}{4} + \frac{- 16}{4}) o = (4 o) - 4 o$

Combine the fractions:

$\frac{(7 - 16)}{4} o = (4 o) - 4 o$

Combine the numerators:

$\frac{- 9}{4} o = (4 o) - 4 o$

Simplify the arithmetic:

$\frac{- 9}{4} o = 0$

Divide both sides by the coefficient:

$o = 0$

8 additional steps

$7 \cdot \frac{1}{4} o = - (4 o)$

Multiply the coefficients:

$\frac{(7 \cdot 1)}{4} o = - (4 o)$

Combine like terms:

$\frac{7}{4} o = - (4 o)$

Add to both sides:

$(\frac{7}{4} o) + 4 o = (- 4 o) + 4 o$

Group the coefficients:

$(\frac{7}{4} + 4) o = (- 4 o) + 4 o$

Convert the integer into a fraction:

$(\frac{7}{4} + \frac{16}{4}) o = (- 4 o) + 4 o$

Combine the fractions:

$\frac{(7 + 16)}{4} o = (- 4 o) + 4 o$

Combine the numerators:

$\frac{23}{4} o = (- 4 o) + 4 o$

Simplify the arithmetic:

$\frac{23}{4} o = 0$

Divide both sides by the coefficient:

$o = 0$

3. List the solutions

$o = 0, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 7 | \frac{1}{4} o |$
$y = | 4 o |$
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 without absolute value bars

2. Solve the two equations for o

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for o

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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