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Solution - Absolute value equations

Exact form:

f = 7

f=7

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
$| f - 6 | = | f - 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| f - 6 \| = \| f - 8 \|$
$x = + y$	$(f - 6) = (f - 8)$
$x = - y$	$(f - 6) = - (f - 8)$
$+ x = y$	$(f - 6) = (f - 8)$
$- x = y$	$- (f - 6) = (f - 8)$

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:

$\| x \| = \| y \|$	$\| f - 6 \| = \| f - 8 \|$
$x = + y$ , $+ x = y$	$(f - 6) = (f - 8)$
$x = - y$ , $- x = y$	$(f - 6) = - (f - 8)$

2. Solve the two equations for f

5 additional steps

$(f - 6) = (f - 8)$

Subtract from both sides:

$(f - 6) - f = (f - 8) - f$

Group like terms:

$(f - f) - 6 = (f - 8) - f$

Simplify the arithmetic:

$- 6 = (f - 8) - f$

Group like terms:

$- 6 = (f - f) - 8$

Simplify the arithmetic:

$- 6 = - 8$

The statement is false:

$- 6 = - 8$

The equation is false so it has no solution.

12 additional steps

$(f - 6) = - (f - 8)$

Expand the parentheses:

$(f - 6) = - f + 8$

Add to both sides:

$(f - 6) + f = (- f + 8) + f$

Group like terms:

$(f + f) - 6 = (- f + 8) + f$

Simplify the arithmetic:

$2 f - 6 = (- f + 8) + f$

Group like terms:

$2 f - 6 = (- f + f) + 8$

Simplify the arithmetic:

$2 f - 6 = 8$

Add to both sides:

$(2 f - 6) + 6 = 8 + 6$

Simplify the arithmetic:

$2 f = 8 + 6$

Simplify the arithmetic:

$2 f = 14$

Divide both sides by :

$\frac{(2 f)}{2} = \frac{14}{2}$

Simplify the fraction:

$f = \frac{14}{2}$

Find the greatest common factor of the numerator and denominator:

$f = \frac{(7 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$f = 7$

3. Graph

Each line represents the function of one side of the equation:
$y = | f - 6 |$
$y = | f - 8 |$
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 f

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 f

3. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved