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Solution - Absolute value equations

Exact form:

h = - \frac{1}{9}, \frac{1}{15}

h=-\frac{1}{9} , \frac{1}{15}

Decimal form:

h = - 0.111, 0.067

h=-0.111 , 0.067

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
$\frac{1}{4} | 3 h - 1 | = | 3 h |$
without the absolute value bars:

$\| x \| = \| y \|$	$\frac{1}{4} \| 3 h - 1 \| = \| 3 h \|$
$x = + y$	$\frac{1}{4} (3 h - 1) = (3 h)$
$x = - y$	$\frac{1}{4} (3 h - 1) = - (3 h)$
$+ x = y$	$\frac{1}{4} (3 h - 1) = (3 h)$
$- x = y$	$\frac{1}{4} (- (3 h - 1)) = (3 h)$

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:

$\| x \| = \| y \|$	$\frac{1}{4} \| 3 h - 1 \| = \| 3 h \|$
$x = + y$ , $+ x = y$	$\frac{1}{4} (3 h - 1) = (3 h)$
$x = - y$ , $- x = y$	$\frac{1}{4} (3 h - 1) = - (3 h)$

2. Solve the two equations for h

23 additional steps

$\frac{1}{4} \cdot (3 h - 1) = 3 h$

Multiply the fraction(s):

$\frac{(1 \cdot (3 h - 1))}{4} = 3 h$

Break up the fraction:

$\frac{3 h}{4} + \frac{- 1}{4} = 3 h$

Subtract from both sides:

$(\frac{3 h}{4} + \frac{- 1}{4}) - 3 h = (3 h) - 3 h$

Group like terms:

$(\frac{3 h}{4} - 3 h) + \frac{- 1}{4} = (3 h) - 3 h$

Group the coefficients:

$(\frac{3}{4} - 3) h + \frac{- 1}{4} = (3 h) - 3 h$

Convert the integer into a fraction:

$(\frac{3}{4} + \frac{- 12}{4}) h + \frac{- 1}{4} = (3 h) - 3 h$

Combine the fractions:

$\frac{(3 - 12)}{4} h + \frac{- 1}{4} = (3 h) - 3 h$

Combine the numerators:

$\frac{- 9}{4} h + \frac{- 1}{4} = (3 h) - 3 h$

Simplify the arithmetic:

$\frac{- 9}{4} h + \frac{- 1}{4} = 0$

Add to both sides:

$(\frac{- 9}{4} h + \frac{- 1}{4}) + \frac{1}{4} = 0 + \frac{1}{4}$

Combine the fractions:

$\frac{- 9}{4} h + \frac{(- 1 + 1)}{4} = 0 + \frac{1}{4}$

Combine the numerators:

$\frac{- 9}{4} h + \frac{0}{4} = 0 + \frac{1}{4}$

Reduce the zero numerator:

$\frac{- 9}{4} h + 0 = 0 + \frac{1}{4}$

Simplify the arithmetic:

$\frac{- 9}{4} h = 0 + \frac{1}{4}$

Simplify the arithmetic:

$\frac{- 9}{4} h = \frac{1}{4}$

Multiply both sides by inverse fraction :

$(\frac{- 9}{4} h) \cdot \frac{4}{- 9} = (\frac{1}{4}) \cdot \frac{4}{- 9}$

Move the negative sign from the denominator to the numerator:

$\frac{- 9}{4} h \cdot \frac{- 4}{9} = (\frac{1}{4}) \cdot \frac{4}{- 9}$

Group like terms:

$(\frac{- 9}{4} \cdot \frac{- 4}{9}) h = (\frac{1}{4}) \cdot \frac{4}{- 9}$

Multiply the coefficients:

$\frac{(- 9 \cdot - 4)}{(4 \cdot 9)} h = (\frac{1}{4}) \cdot \frac{4}{- 9}$

Simplify the arithmetic:

$1 h = (\frac{1}{4}) \cdot \frac{4}{- 9}$

$h = (\frac{1}{4}) \cdot \frac{4}{- 9}$

Move the negative sign from the denominator to the numerator:

$h = \frac{1}{4} \cdot \frac{- 4}{9}$

Multiply the fraction(s):

$h = \frac{(1 \cdot - 4)}{(4 \cdot 9)}$

Simplify the arithmetic:

$h = \frac{- 1}{9}$

20 additional steps

$\frac{1}{4} \cdot (3 h - 1) = - (3 h)$

Multiply the fraction(s):

$\frac{(1 \cdot (3 h - 1))}{4} = - (3 h)$

Break up the fraction:

$\frac{3 h}{4} + \frac{- 1}{4} = - (3 h)$

Add to both sides:

$(\frac{3 h}{4} + \frac{- 1}{4}) + 3 h = (- 3 h) + 3 h$

Group like terms:

$(\frac{3 h}{4} + 3 h) + \frac{- 1}{4} = (- 3 h) + 3 h$

Group the coefficients:

$(\frac{3}{4} + 3) h + \frac{- 1}{4} = (- 3 h) + 3 h$

Convert the integer into a fraction:

$(\frac{3}{4} + \frac{12}{4}) h + \frac{- 1}{4} = (- 3 h) + 3 h$

Combine the fractions:

$\frac{(3 + 12)}{4} h + \frac{- 1}{4} = (- 3 h) + 3 h$

Combine the numerators:

$\frac{15}{4} h + \frac{- 1}{4} = (- 3 h) + 3 h$

Simplify the arithmetic:

$\frac{15}{4} h + \frac{- 1}{4} = 0$

Add to both sides:

$(\frac{15}{4} h + \frac{- 1}{4}) + \frac{1}{4} = 0 + \frac{1}{4}$

Combine the fractions:

$\frac{15}{4} h + \frac{(- 1 + 1)}{4} = 0 + \frac{1}{4}$

Combine the numerators:

$\frac{15}{4} h + \frac{0}{4} = 0 + \frac{1}{4}$

Reduce the zero numerator:

$\frac{15}{4} h + 0 = 0 + \frac{1}{4}$

Simplify the arithmetic:

$\frac{15}{4} h = 0 + \frac{1}{4}$

Simplify the arithmetic:

$\frac{15}{4} h = \frac{1}{4}$

Multiply both sides by inverse fraction :

$(\frac{15}{4} h) \cdot \frac{4}{15} = (\frac{1}{4}) \cdot \frac{4}{15}$

Group like terms:

$(\frac{15}{4} \cdot \frac{4}{15}) h = (\frac{1}{4}) \cdot \frac{4}{15}$

Multiply the coefficients:

$\frac{(15 \cdot 4)}{(4 \cdot 15)} h = (\frac{1}{4}) \cdot \frac{4}{15}$

Simplify the fraction:

$h = (\frac{1}{4}) \cdot \frac{4}{15}$

Multiply the fraction(s):

$h = \frac{(1 \cdot 4)}{(4 \cdot 15)}$

Simplify the arithmetic:

$h = \frac{1}{15}$

3. List the solutions

$h = - \frac{1}{9}, \frac{1}{15}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = \frac{1}{4} | 3 h - 1 |$
$y = | 3 h |$
The equation is true where the two lines cross.

How did we do?

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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 h

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 h

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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