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Solution - Absolute value equations

Exact form:

h = - 1, \frac{1}{5}

h=-1 , \frac{1}{5}

Decimal form:

h = - 1, 0.2

h=-1 , 0.2

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
$0.4 | 10 h - 5 | = | 6 h |$
without the absolute value bars:

$\| x \| = \| y \|$	$0.4 \| 10 h - 5 \| = \| 6 h \|$
$x = + y$	$0.4 (10 h - 5) = (6 h)$
$x = - y$	$0.4 (10 h - 5) = - (6 h)$
$+ x = y$	$0.4 (10 h - 5) = (6 h)$
$- x = y$	$0.4 (- (10 h - 5)) = (6 h)$

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:

$\| x \| = \| y \|$	$0.4 \| 10 h - 5 \| = \| 6 h \|$
$x = + y$ , $+ x = y$	$0.4 (10 h - 5) = (6 h)$
$x = - y$ , $- x = y$	$0.4 (10 h - 5) = - (6 h)$

2. Solve the two equations for h

14 additional steps

$0.4 \cdot (10 h - 5) = 6 h$

Expand the parentheses:

$0.4 \cdot 10 h + 0.4 \cdot - 5 = 6 h$

Multiply the coefficients:

$4 h + 0.4 \cdot - 5 = 6 h$

Simplify the arithmetic:

$4 h - 2 = 6 h$

Subtract from both sides:

$(4 h - 2) - 6 h = (6 h) - 6 h$

Group like terms:

$(4 h - 6 h) - 2 = (6 h) - 6 h$

Simplify the arithmetic:

$- 2 h - 2 = (6 h) - 6 h$

Simplify the arithmetic:

$- 2 h - 2 = 0$

Add to both sides:

$(- 2 h - 2) + 2 = 0 + 2$

Simplify the arithmetic:

$- 2 h = 0 + 2$

Simplify the arithmetic:

$- 2 h = 2$

Divide both sides by :

$\frac{(- 2 h)}{- 2} = \frac{2}{- 2}$

Cancel out the negatives:

$\frac{2 h}{2} = \frac{2}{- 2}$

Simplify the fraction:

$h = \frac{2}{- 2}$

Move the negative sign from the denominator to the numerator:

$h = \frac{- 2}{2}$

Simplify the fraction:

$h = - 1$

13 additional steps

$0.4 \cdot (10 h - 5) = - (6 h)$

Expand the parentheses:

$0.4 \cdot 10 h + 0.4 \cdot - 5 = - (6 h)$

Multiply the coefficients:

$4 h + 0.4 \cdot - 5 = - (6 h)$

Simplify the arithmetic:

$4 h - 2 = - (6 h)$

Add to both sides:

$(4 h - 2) + 6 h = (- 6 h) + 6 h$

Group like terms:

$(4 h + 6 h) - 2 = (- 6 h) + 6 h$

Simplify the arithmetic:

$10 h - 2 = (- 6 h) + 6 h$

Simplify the arithmetic:

$10 h - 2 = 0$

Add to both sides:

$(10 h - 2) + 2 = 0 + 2$

Simplify the arithmetic:

$10 h = 0 + 2$

Simplify the arithmetic:

$10 h = 2$

Divide both sides by :

$\frac{(10 h)}{10} = \frac{2}{10}$

Simplify the fraction:

$h = \frac{2}{10}$

Find the greatest common factor of the numerator and denominator:

$h = \frac{(1 \cdot 2)}{(5 \cdot 2)}$

Factor out and cancel the greatest common factor:

$h = \frac{1}{5}$

3. List the solutions

$h = - 1, \frac{1}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 0.4 | 10 h - 5 |$
$y = | 6 h |$
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 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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