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Solution - Absolute value equations

Exact form:

h = - \frac{1}{70}, \frac{1}{76}

h=-\frac{1}{70} , \frac{1}{76}

Decimal form:

h = - 0.014, 0.013

h=-0.014 , 0.013

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
$| 3 h - 1 | = | 73 h |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 h - 1 \| = \| 73 h \|$
$x = + y$	$(3 h - 1) = (73 h)$
$x = - y$	$(3 h - 1) = - (73 h)$
$+ x = y$	$(3 h - 1) = (73 h)$
$- x = y$	$- (3 h - 1) = (73 h)$

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:

$\| x \| = \| y \|$	$\| 3 h - 1 \| = \| 73 h \|$
$x = + y$ , $+ x = y$	$(3 h - 1) = (73 h)$
$x = - y$ , $- x = y$	$(3 h - 1) = - (73 h)$

2. Solve the two equations for h

10 additional steps

$(3 h - 1) = 73 h$

Subtract from both sides:

$(3 h - 1) - 73 h = (73 h) - 73 h$

Group like terms:

$(3 h - 73 h) - 1 = (73 h) - 73 h$

Simplify the arithmetic:

$- 70 h - 1 = (73 h) - 73 h$

Simplify the arithmetic:

$- 70 h - 1 = 0$

Add to both sides:

$(- 70 h - 1) + 1 = 0 + 1$

Simplify the arithmetic:

$- 70 h = 0 + 1$

Simplify the arithmetic:

$- 70 h = 1$

Divide both sides by :

$\frac{(- 70 h)}{- 70} = \frac{1}{- 70}$

Cancel out the negatives:

$\frac{70 h}{70} = \frac{1}{- 70}$

Simplify the fraction:

$h = \frac{1}{- 70}$

Move the negative sign from the denominator to the numerator:

$h = \frac{- 1}{70}$

7 additional steps

$(3 h - 1) = - 73 h$

Add to both sides:

$(3 h - 1) + 1 = (- 73 h) + 1$

Simplify the arithmetic:

$3 h = (- 73 h) + 1$

Add to both sides:

$(3 h) + 73 h = ((- 73 h) + 1) + 73 h$

Simplify the arithmetic:

$76 h = ((- 73 h) + 1) + 73 h$

Group like terms:

$76 h = (- 73 h + 73 h) + 1$

Simplify the arithmetic:

$76 h = 1$

Divide both sides by :

$\frac{(76 h)}{76} = \frac{1}{76}$

Simplify the fraction:

$h = \frac{1}{76}$

3. List the solutions

$h = - \frac{1}{70}, \frac{1}{76}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 h - 1 |$
$y = | 73 h |$
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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