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Solution - Absolute value equations

Exact form:

r = \frac{7}{16}

r=\frac{7}{16}

Decimal form:

r = 0.438

r=0.438

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
$| r | = | r - \frac{7}{8} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| r \| = \| r - \frac{7}{8} \|$
$x = + y$	$(r) = (r - \frac{7}{8})$
$x = - y$	$(r) = - (r - \frac{7}{8})$
$+ x = y$	$(r) = (r - \frac{7}{8})$
$- x = y$	$- (r) = (r - \frac{7}{8})$

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:

$\| x \| = \| y \|$	$\| r \| = \| r - \frac{7}{8} \|$
$x = + y$ , $+ x = y$	$(r) = (r - \frac{7}{8})$
$x = - y$ , $- x = y$	$(r) = - (r - \frac{7}{8})$

2. Solve the two equations for r

4 additional steps

$r = (r + \frac{- 7}{8})$

Subtract from both sides:

$r - r = (r + \frac{- 7}{8}) - r$

Simplify the arithmetic:

$0 = (r + \frac{- 7}{8}) - r$

Group like terms:

$0 = (r - r) + \frac{- 7}{8}$

Simplify the arithmetic:

$0 = \frac{- 7}{8}$

The statement is false:

$0 = \frac{- 7}{8}$

The equation is false so it has no solution.

8 additional steps

$r = - (r + \frac{- 7}{8})$

Expand the parentheses:

$r = - r + \frac{7}{8}$

Add to both sides:

$r + r = (- r + \frac{7}{8}) + r$

Simplify the arithmetic:

$2 r = (- r + \frac{7}{8}) + r$

Group like terms:

$2 r = (- r + r) + \frac{7}{8}$

Simplify the arithmetic:

$2 r = \frac{7}{8}$

Divide both sides by :

$\frac{(2 r)}{2} = \frac{(\frac{7}{8})}{2}$

Simplify the fraction:

$r = \frac{(\frac{7}{8})}{2}$

Simplify the arithmetic:

$r = \frac{7}{(8 \cdot 2)}$

$r = \frac{7}{16}$

3. Graph

Each line represents the function of one side of the equation:
$y = | r |$
$y = | r - \frac{7}{8} |$
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 r

3. 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 r

3. Graph

Why learn this

Terms and topics

Related links

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