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Solution - Absolute value equations

Exact form:

h = 4

h=4

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
$| h - 8 | = | h |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| h - 8 \| = \| h \|$
$x = + y$	$(h - 8) = (h)$
$x = - y$	$(h - 8) = - (h)$
$+ x = y$	$(h - 8) = (h)$
$- x = y$	$- (h - 8) = (h)$

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:

$\| x \| = \| y \|$	$\| h - 8 \| = \| h \|$
$x = + y$ , $+ x = y$	$(h - 8) = (h)$
$x = - y$ , $- x = y$	$(h - 8) = - (h)$

2. Solve the two equations for h

4 additional steps

$(h - 8) = h$

Subtract from both sides:

$(h - 8) - h = h - h$

Group like terms:

$(h - h) - 8 = h - h$

Simplify the arithmetic:

$- 8 = h - h$

Simplify the arithmetic:

$- 8 = 0$

The statement is false:

$- 8 = 0$

The equation is false so it has no solution.

10 additional steps

$(h - 8) = - h$

Add to both sides:

$(h - 8) + h = - h + h$

Group like terms:

$(h + h) - 8 = - h + h$

Simplify the arithmetic:

$2 h - 8 = - h + h$

Simplify the arithmetic:

$2 h - 8 = 0$

Add to both sides:

$(2 h - 8) + 8 = 0 + 8$

Simplify the arithmetic:

$2 h = 0 + 8$

Simplify the arithmetic:

$2 h = 8$

Divide both sides by :

$\frac{(2 h)}{2} = \frac{8}{2}$

Simplify the fraction:

$h = \frac{8}{2}$

Find the greatest common factor of the numerator and denominator:

$h = \frac{(4 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$h = 4$

3. Graph

Each line represents the function of one side of the equation:
$y = | h - 8 |$
$y = | h |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved